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School of Mechanical Engineering,
Faculty of Engineering, Tel-Aviv
University, Tel-Aviv, Israel
e-mail: gelfgat@tau.ac.il
To appear in “Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics”, ed.
A. Gelfgat, Springer, 2018.
Global Galerkin method for stability studies in incompressible CFD and other
possible applications
Alexander Gelfgat
Abstract In this paper the author reviews a version of the global Galerkin that was developed
and applied in a series of earlier publications. The method is based on divergence-free basis
functions satisfying all the linear and homogeneous boundary conditions. The functions are
defined as linear superpositions of the Chebyshev polynomials of the first and second klinds that
are combined into divergence free vectors. The description and explanations of treatment of
boundary conditions inhomogeneities and singularities are given. Possible implementation for
steady state solvers, path-continuation, stability solvers and straight-forward integration in time
are discussed. The most important results obtained using the approach are briefly reviewed and
possible future applications are discussed.
Keywords: weighted residual methods, spectral methods, Galerkin method, instability, path-
continuation
1 Introduction
This paper revisits a version of the global Galerkin method whose development started in
the author’s PhD thesis (Gelfgat, 1988) and resulted in an effective approach for stability
analysis of model incompressible flows, so that well known and less known results had been
2
published continuously between 1994 and 2005. Several years ago I was asked to present this
approach again, and it triggered some interest among several young colleagues. It is mainly their
interest that inspired me to review all that was done and to make a consistent description of all
the technical details, which were distributed over several papers, and sometimes omitted.
Naturally, I have added some comments on my personal current opinion of what was done and
what possibly can be done in the future.
The development of this approach started in the era when computer memory was
measured in kilobytes, so that even storage of a large matrix was a problem. The only way to
reduce the required memory was to reduce the number of degrees of freedom. Since
discretization of the flow region (domain) requires fine grids, the decrease of number of degrees
of freedom was sought in different versions of global weighted residual methods, called also
spectral methods. These methods do not discretize the domain, and approximate solutions as
truncated series of basis functions. It was shown by many authors that weighted residuals
methods allow for a very significant reduction of degrees of freedom (see, e.g., Canuto et al.,
2006). Straight-forward application of these methods to incompressible fluid dynamics involves
algebraic constraints that are related to the boundary conditions and the continuity equation.
Removal of these constraints by an appropriate choice of basis functions would decrease the
number of degrees of freedom even more. The construction of such basis functions, is the main
topic of this text. For problems with periodic boundary conditions the choice of Fourier series is
natural, and also allows for application of the fast Fourier transform when the non-linear terms
are evaluated by the pseudo-spectral approach. However, the choice of basis functions is not so
obvious for non-periodic boundary conditions. An obvious idea to linearly combine well-known
functions, e.g., the Chebyshev polynomials, into expressions that satisfy linear and homogeneous
boundary conditions of a problem appears in Orszag (1971a,b). To the best of the author’s
knowledge, this is the first appearance of this idea, at least in CFD. Many authors, including this
one, rediscovered this way of constructing the basis functions (see Section 9).
The next step was to combine these linear superpositions into a two-dimensional
divergence-free vector, for which Chebyshev polynomials of the first and second kinds fit very
well (Gelfgat 1988, Gelfgat & Tanasawa, 1994). The Galerkin method based on divergence-free
basis functions that satisfy all linear and homogeneous boundary conditions (LHBC in the
further text), led to a noticeable reduction of the number of degrees of freedom, as was already
3
shown in Gelfgat & Tanasawa (1994). Later, as computational power grew, so did the
possibilities of applications of the method. A small number of degrees of freedom resulted in
Jacobian matrices of small size, which could be treated numerically for the computation of
steady flows, as well as for the solution of the eigenvalue problems associated with linear
stability (Gelfgat et al., 1997, 1999a,b). Divergence free functions were extended to cylindrical
coordinates, which allowed us to consider flow in the rotating disk – cylinder system and to
obtain the first stability results (Gelfgat et al., 1996) for this configuration.
Since then a number of different problems addressing steady flows, multiplicity of states,
and stability were solved for different flows in two-dimensional rectangular cavities and
cylinders. The periodic circumferential coordinate in the cylindrical geometry allows one to
study stability of an axisymmetric base flow with respect to three-dimensional perturbations. The
linearized problem for the 3D disturbances separates for each circumferential Fourier mode, so
that the final answer is obtained by consideration of several 2D-like problems. Subsequently, a
considerable number of results were obtained for stability of flows in cylinders, which were
driven by the rotation of boundaries, by buoyancy convection, and by magnetic field.
The first attempt to study 3D instability in Cartesian coordinates was carried out in
Gelfgat (1999) for the Rayleigh-Bénard problem in a rectangular box, i.e., the stability of a
quiescent fluid heated from below. No attempts were made by the author to study stability of
fully developed 3D flow. At the same time, the three-dimensional bases found an unexpected
application for visualization of incompressible 3D flows (Gelfgat, 2014).
In what follows we start from a brief description of the weighted residual and Galerkin
method formulated for an incompressible fluid dynamics problem. Then the proposed way of
constructing basis functions is explained, starting from a one-dimensional two-point problem.
The treatment of inhomogeneities and boundary conditions singularities is explained. This
follows by discussion of the resulting dynamical system and explanations of how it was treated
in the cited works, as well as how it can be treated for fully 3D problems. After that, several
illustrative examples are presented. Finally, a discussion of possible future studies is given.
2 The problem and the numerical method
We consider flow of an incompressible fluid in a two-dimensional rectangle 0
, 0 , or in a three-dimensional box 0, 0 , 0 . The
4
rectangular shape of the domain is the main restriction for all the following. Below we discuss
how this restriction can be relaxed to a canonical shape, i.e., the domain bounded by the
coordinate surfaces belonging to a system of orthogonal curvilinear coordinates. The momentum
and continuity equations for velocity and pressure read
+()=+
+ , (2.1)
= 0 , (2.2)
where is the Reynolds number. We assume also that boundary conditions for all three
velocity components are linear and homogeneous, e.g., the no-slip conditions on all boundaries.
Equations (2.1)-(2.2) can be considered together with other scalar transport equations for, e.g.,
temperature and/or concentration, an example of which will be given below.
To formulate the Galerkin method we assume that the solution belongs to a space V of
divergence-free vectors satisfying all the (linear and homogeneous) boundary conditions, LHBC.
Assume that vectors {}
form a basis in this space. Then the solution can be represented
as =
. (2.3)
The coefficients can be obtained by evaluation of inner products of Eq. (2.3) with a basis
vector : ,=,
, = 1,2,3, … (2.4)
Here we assumed that the space V is supplied with an inner product ,, which is yet to be
defined. If the functions {}
form an orthogonal basis, then the coefficients are obtained
as =,
,, = 1,2,3, … (2.5)
However, if the basis functions are not orthogonal, the expressions (2.4) form an infinite system
of linear algebraic equations, which can be solved only up to a certain truncation. Keeping the
first terms in the series (2.3) and defining a vector of coefficients ={,,, … , }, the
first coefficients can be obtained as
=, =,, =, . (2.6)
Here is the Gram matrix, and is the truncation number.
5
In the relations (2.4)-(2.6) we assumed that the vector is known. If it is unknown, the
coefficients can be obtained only approximately by minimization of the residual of the
momentum equation. To show how they can be calculated, we first assume that the coefficients
are time-dependent and the basis functions {}
depend only on coordinate values. Then
the representation (2.3) defines a time- and space-dependent function . Note that the continuity
equation (2.2), as well as all the boundary conditions are already satisfied because belongs to
the space V. Clearly, the solution we are looking for also belongs to this space, so that the
momentum equation is the only one to be solved. Now we rewrite the momentum equation (2.1)
in two additional and equivalent forms
+()+
== (2.7)
=+
()+ (2.8)
Here is the residual of the momentum equation. If is the solution then 0. For a general
case we assume that all possible residuals belong to a certain functional space W, which is also
supplied with an inner product , . Assume also that {}
is a basis in W. Then the
requirement that the residual be zero is equivalent to the requirement that the residual is
orthogonal to all the basis functions , namely
,= 0, = 1, 2, 3, … (2.9)
The equations (2.9) form an infinite set of non-linear time-dependent ODEs that must be solved
to find the time-dependent coefficients (). However, this system of equations is not closed
yet, since nothing is said about the pressure. To proceed, we assume that the pressure belongs to
a space S of scalar time-dependent functions, differentiable at least twice in the domain, with the
basis {}
. The pressure is represented as
=()
. (2.10)
The equation for pressure is formed in the standard way, by applying the divergence operator to
both sides of the momentum equation (2.1), which yields
=[()] . (2.11)
The boundary conditions required this equation will be discussed later. Note, that the velocity
representation (2.3), even in the truncated form used below, guarantees zero velocity divergence.
6
Since the Laplacian and divergence operators are evaluated analytically, they commute in every
approximate (truncated) formulation. We introduce the residual of the pressure equation
=+[()]. (2.12)
For the residual we demand only piecewise continuity in all spatial directions, so that it
formally belongs to another space of scalar functions. This space is denoted as D, its basis as
{}
, and the scalar product as ,. Now, for the solution of the problem represented by
and , the scalar non-linear algebraic equations
,= 0, = 1, 2, 3, … (2.13)
must be satisfied. Equations (2.13) define the coefficients () and must be satisfied together
with the equations (2.9). Note that these equations do not contain the time derivative and,
therefore, are algebraic constraints for the ODEs (2.9).
Obviously, VW and SD. To build a numerical procedure for obtaining the first
and coefficients of the velocity and pressure series (2.3) and (2.10), we truncate the series
together with the residual projection equations (2.9) and (2.13). This results in
()
, ,= 0, = 1, 2, 3, … , , (2.14)
()
, ,= 0, = 1, 2, 3, … , , (2.15)
which defines non-linear ODEs and non-linear algebraic constraints for calculation of the
time-dependent coefficients () and (). This is called method of weighted residuals
(Fletcher (1984), Boyd (2000), Canuto et al. (2006)). The ODEs and the algebraic constraints
resulting from projections of the momentum and pressure equation residuals onto the basis
functions are
(),
= ()()(),
()
,+
(),+
, , = 1, 2, 3, … , (2.16)
()
,= ()()(),+,
(2.17)
= 1, 2, 3, … ,
Following common definitions, {}
and {}
are called coordinate basis systems, while
{}
and {}
- are called projection basis systems.
7
Note that the weighted residuals method can be formulated also for coordinate basis
functions that do not satisfy some or all boundary conditions. Fletcher (1984) distinguishes
between coordinate functions that satisfy only a (linear) differential equation, satisfy only
boundary conditions, and do not satisfy anything, calling these three cases boundary, interior,
and mixed, respectively. Within this classification, and noting that the equations are non-linear,
we discuss mainly interior methods.
To arrive at the Galerkin formulation we assume that the boundary conditions do not
explicitly involve time. Then the l.h.s. of Eq. (2.8) satisfies the LHBC of the velocity and is
divergence-free. Then, the same can be said about the r.h.s. of Eq. (2.8), and finally about the
residual of the momentum equation defined in Eq. (2.7). Thus, for time-independent boundary
conditions, the residual belongs to the space V, so that we can choose = for all .
Also assuming that the coordinate systems {}
and {}
are usually constructed from
trigonometric functions or polynomials that are infinitely differentiable, the residual will also
be differentiable infinite number of times. Since the physical problem does not specify any
pressure boundary conditions, we can assume that both and belong to the same space S, and
we can choose = for all . Thus, we arrive at a particular version of the weighted
residuals method, in which the coordinate and projection basis systems coincide. This version is
known as the Galerkin or Boubnov – Galerkin method.
An obvious reason to choose the Galerkin method among all possible weighted residual
formulations follows from the fact that VW. By projecting onto a smaller space V, we expect
that with increase of the truncation number convergence will be faster. Another, less obvious and
more profound reason follows from the definition of the inner products via volume integrals. For
scalar and vector functions defined in the domain and on its boundary we define
(,,),(,,)=(,,)
, (2.18)
(,,),(,,)=(,,)
, (2.19)
where (,,)> 0 is the weight function. The simplest and the most robust formulation is
obtained with the unit weight (,,)= 1, for which the inner products (2,18), (2.19) may also
have some physical meaning. For example, the norm produced by (2.19) becomes twice the
dimensionless kinetic energy. An important additional advantage of the unit weight follows from
consideration of the inner product of the gradient of a scalar field (,,) with a divergence
8
free vector field (,,). Assume that is the domain boundary, its normal, and that the
component of normal to the boundary vanishes. Keeping in mind that = 0 and = 0,
we have
,=
=[()]
=()
=
= 0 (2.20)
Thus, if the velocity does not penetrate the boundary, and the inner product is chosen as in
(2.19), the projection of the pressure gradient on all the velocity basis functions vanishes. This
means that equation systems (2.16) and (2.17) separate, so that velocity can be calculated from
(2.14) without any knowledge about pressure. Note that this calculation involves both
minimization of the residual and summation of the truncated velocity series. The pressure then
can be computed from (2.15) using the previously found velocity field. In this case Eqs. (2.16),
which minimizes the residual by computing the time-dependent coefficients (), become
(),
=
()()(),
+
(),+
, (2.21)
= 1, 2, 3, … ,
This is the formal Galerkin formulation for calculating an approximate solution of a problem.
Note that the exclusion of pressure by the Galerkin projection (2.20) is not restricted to closed
regions with non-penetrative boundaries. The inhomogeneity in the velocity boundary conditions
can be removed by change of variables, which is discussed below in more detail.
To proceed we need to explain how to build basis functions, which are divergence-free
and satisfy the whole set of LHBC. Then we will discuss how to handle inhomogeneous
boundary conditions, curvilinear coordinates, and weight functions others than unity.
3 Basis functions
We start from the question of how to satisfy all the LHBC for a scalar unknown function.
We assume that the basis functions for all three-dimensional time-dependent scalar variables,
e.g., temperature and/or concentration, are represented as products of some one-dimensional
bases. Thus for a function (,,,) defined in a box 0, 0 , 0 we
seek a representation in the form of the tensor (Kronecker) product of one-dimensional bases
9
(,,,)= ()()()()
, (3.1)
Here () are unknown time-dependent coefficients, ,, are the truncation numbers
specified in each spatial direction separately, and (),(),() are one-dimensional bases
that must be defined in each direction. The three-dimensional basis corresponding to those
defined in the previous chapter is
(,,)=()()(), =(1)+1+ (3.2)
Starting from here we will use capital letters for the global indices, and small letters for the one-
dimensional indices.
If all the boundary conditions for are linear and homogeneous, the functions
(),() and () must satisfy the boundary conditions posed in the x-, y-, and z-
directions, respectively. Assume that there are M boundary conditions in the x-direction posed in
the form ()()
= 0, = 1, 2, … , . (3.3)
Here are known coefficients, is the derivative number, and are the borders (that are
parts of coordinate surfaces =) where the boundary conditions are posed. Note that m is the
number of a boundary condition and not a number of the point, since several boundary
conditions can be defined at the same point, so that sometimes = for . For the
following we can also extend (3.3) by assuming that negative correspond to integrals, and that
surfaces = are not necessarily the boundary points, but also can lie inside the flow region,
as it happens in a two-fluid example below. Now assume that a set of functions {()}
forms a basis in, say, ([0, ]). This can be, for example, a trigonometric Fourier basis, or a
set of orthogonal polynomials defined on [0, ]. For some reasons we choose this basis for
representation of the solution, but the functions () do not satisfy the boundary conditions
(3.3). We build an alternative basis by considering superpositions of + 1 consequent basis
functions as ()=()
(3.4)
Substituting () into the boundary conditions (3.3) we obtain
()
=
()()
= 0, = 1, 2, … , (3.5)
10
For each the relations (3.5) form a system of M equations that can be used to find + 1
coefficients . To make the equations solvable we choose ,= 1, that leaves us with M
linear algebraic equations for M remaining coefficients for every fixed . The matrix of this
system will be regular if the boundary conditions (3.3) are independent. In this way, the
functions () are fully defined and satisfy all the boundary conditions (3.3). Obviously they
form a basis in the subspace {()} ([0, ]), which contains functions that satisfy
conditions (3.3) and are differentiable infinite number of times. In this way we form bases
(),() and () for the representation (3.1).
In the following all the examples will be based on the Chebyshev polynomials of the first
and the second type, () and (), briefly described in the Appendix A. In the further text
these polynomials will always be chosen as the basis {()}
.
3.1. Example: Two-point boundary value problem
Consider a two-point boundary value problem for () posed on the interval 01
with the two (= 2) boundary conditions
(0)+ (0)= 0, (1)+ (1)= 0, (3.1.1)
where ,,, are known coefficients. Recalling the Chebyshev polynomials and taking
into account ,= 1, our new basis functions are defined as
()=()
=()+()+() . (3.1.2)
The boundary values of the Chebyshev polynomials and their derivatives are given in Appendix
A. Substituting () into the boundary conditions (3.1.1) and using Eqs. (A3) and (A5), we
obtain two equations for the coefficients and :
[2(+ 1)+]+[2(+ 2)+]=2 (3.1.3)
[2(+ 1)+]+[2(+ 2)+]=2 (3.1.4)
These equation are easily solved analytically, which yields
=()()
[()][()][()][()] , (3.1.5)
=()()
[()][()][()][()] , (3.1.6)
11
and defines a new basis, whose satisfy both boundary conditions. This idea was introduced in
Orszag (1971a) for two-point homogeneous Dirichlet boundary conditions, and in Orszag
(1971b) for boundary conditions of the Orr-Sommerfeld equation. It was formalized for an
arbitrary set of boundary conditions (3.1.1) in Gelfgat (1988). Note that for a similar problem
defined on the interval 0 we need only to replace by
in Eq. (3.1.2), which will
not affect the expressions (3.1.5) and (3.1.6).
3.2. Two-dimensional divergence-free basis
To construct basis functions, which are two-dimensional divergence-free vectors
satisfying all the boundary conditions, we start by constructing a divergence-free basis that does
not yet involve any boundary conditions. Using the Chebyshev polynomials, and assuming the
flow region is a rectangle 0, 0 , we define
=
()
()=
, ,= 1,2,3, … (3.2.1)
=
0, =0
(3.2.2)
Applying the relation (A2), it is easily seen that = 0. Now, to implement the boundary
conditions and to keep the divergence zero, we keep the x- and y- components of the vector
dependent on each other, as in (3.2.1). To do so we implement all the velocity boundary
conditions in the x-direction in the x-dependent part of the vector, and do the same in the y-
direction. In the x-direction the boundary conditions are posed at = 0 and = for the two
velocity components, so that we have four boundary conditions in total. The same is done in the
y-direction. Thus, we construct the basis using linear superpositions of five (4+1) consecutive
Chebyshev polynomials. This results in
(,)=
(3.2.3)
12
It is easy to check that = 0. The coefficients and must be found by substituting
into the boundary conditions. For example, assume that the rectangle has a stress free
boundary at =, while all the other boundaries are no-slip. This leads to the following
boundary conditions for the velocity :
(= 0, )=(=,)=(,= 0)= 0, ,==
,== 0 (3.2.4)
The assignment =0= 1 and substitution of (3.2.3) into (3.2.4) yields
== 0, =
,=
(3.2.5)
=
()()
()(), =()()
() , > 0 (3.2.6)
=
, = 2
,> 0 (3.2.7)
=
, =2
,> 0 (3.2.8)
=
, =2
,> 0 (3.2.9)
=
, =
,> 0 (3.2.10)
3.3. Three-dimensional divergence-free basis
Generalization of the 2D basis functions for a three-dimensional case is not straight-
forward, since it is unclear how to produce divergence-free vectors, similar to those defined in
(3.2.3), that will form a complete set of basis functions. To use a similar approach, we need to
carry out several additional calculations.
Assume that =(,,) is a divergence-free three-dimensional vector that satisfies the
no-slip conditions on all the boundaries of the rectangular box 0, 0 , 0
. Since =
+
+
= 0, so that =(
+
), a 3D
incompressible vector field can be decomposed as
=
=
0
+0
, =
,
=
(3.3.1)
13
Since both vectors (,)=(,0,) and (,)=(0, ,) are divergence-free, this
decomposition shows that a divergence-free velocity field can be represented as superposition of
two fields which have components only in the (x,z) or (y,z) planes. For each of the two vectors
we can construct basis functions similar to (3.2.3)
(,)(,,)=
()
0
()
(3.3.2)
(,)(,,)=
0
()
()
(3.3.3)
As above, the coefficients , , , , , and are defined after substitution of the functions
in the boundary conditions. Note that the number of polynomials included in the linear superpositions in
z-direction, + 1, is not yet defined. This is because at = only the sum (,,)+
(,,) is zero, while the boundary values of (,,) and (,,) are not known.
An example of such a decomposition can be found in Gelfgat (2014). Therefore there are only
three boundary conditions in the z-direction to be satisfied by the basis functions
(,) and
(,), so that = 3. It is still possible to use these bases, but to satisfy the boundary conditions
for at =, additional algebraic constraints will be needed. Note that there is no such a
problem if boundary conditions in the z-direction are periodic.
To remove the above algebraic constraint at =, we assume that = 4 in (3.3.2) and
(3.3.3), so that the functions
(,) and
(,) are divergence-free and satisfy all the boundary
conditions. In this case, using (3.2.5) and (3.2.6),
======, ====== 0,
==()
(3.3.4)
Projection of the solution onto
(,) and
(,) results in a vector
14
=0
+0
, (3.3.5)
which is divergence-free, satisfies all the boundary conditions, but may not be a good
approximation of because the set of basis functions still is not complete. To complete the basis
we notice that
(,) and
(,) project the velocity on the (,) and (,)
planes, so that it is straight-forward to add projections on the (,) planes as well. Thus,
similarly to (3.3.2) and (3.3.3) we add another set of divergence-free basis functions satisfying
all the boundary conditions
(,)(,,)=
()
()
0
(3.3.6)
Similarly to previous functions, for the no-slip boundary conditions posed on all boundaries,
==, === 0, =()
(3.3.7)
Finally, we seeek a three-dimensional solution of the form
,,
(,)
(,)
(,)
()
(,)(,,)
(,)
+
+ ,,
(,)
(,)
(,)
()
(,)(,,)
(,)
+
(3.3.8)
+ ,,
(,)
(,)
(,)
()
(,)(,,)
(,)
Projection of the residual of the momentum equation on all three sets of basis vectors yields three
sets of ODEs for calculation of the three sets of time-dependent coefficients ,,
(,),,,
(,), and
,,
(,). Since the basis functions are not orthogonal it will be necessary to invert the Gram matrix
that is formed as
15
=
(,),
(,)
(,),
(,)
(,),
(,)
(,),
(,)
(,),
(,)
(,),
(,)
(,),
(,)
(,),
(,)
(,),
(,)
(3.3.9)
A simple numerical test for the no-slip boundary conditions and equal truncation numbers
(starting from 4 and larger) in each direction and for each set of the functions shows that the
Gram matrix is singular. This means that some of the functions are linearly dependent and must
be excluded. Based on the above discussion, we see that in the case of periodic boundary
conditions in the z-direction, all the set (3.3.6) can be omitted. However, some functions of this
set must be added in the case of no-slip boundaries. This shows that the complete set of linearly
independent basis functions differs for different boundary conditions. Unfortunately, the author
could not arrive at a rigorous mathematical procedure that would determine which functions
must be excluded at certain boundary conditions. At the same time, a simple numerical
experiment can be helpful.
Considering no-slip conditions at all the boundaries, we varied (,) in the last sum of
Eq. (3.3.7). In other words, we varied the truncation number in the z-direction for the functions
defined in (3.3.6) only. We found that by taking (,)= 0, i.e., one basis function in the z-
direction, we obtain a regular Gram matrix. Increase of (,) to (,)1, makes the Gram
function singular. Furthermore, taking a single basis function in the z-direction, with the third
index 1, which means polynomials of larger degrees in (3.3.6), also results in a singular
Gram matrix. This shows that the addition of the first polynomial in the z-direction,
corresponding to (,)= 0, is essential, while the others can be omitted. Using (3.3.7), this first
polynomial is 8(). Returning to the sets (3.3.2) and (3.3.3) with the coefficients defined in
(3.3.4), we observe that the polynomials corresponding to the x- and y- vector components start
form the second degree, while those corresponding to the z-component start from the third one.
Thus, the missing second-order polynomial must be added with the help of another set (3.3.6).
3.4. Basis functions in cylindrical and other curvilinear coordinates
16
Consider flow in a cylinder with radius R and height H. The whole problem is defined
now in the cylindrical coordinates (,,), 0 , 0 2, 0 . Using 2-
periodicity in the azimuthal direction we represent the flow as a Fourier series
=(,)()
, (3.4.1)
so that the basis functions in the -direction are complex exponents (). The continuity
equation for (,)=(,),(,),(,) is
()
+
+
= 0 (3.4.2)
Here we must distinguish between the axisymmetric case and axisymmetric Fourier mode, for
which = 0, and all the other 0 modes. Axisymmetric flow (or the axisymmetric mode) is
represented by a single set of the basis functions, which is built similarly to the above 2D
Cartesian case, but taking into account the continuity equation (3.4.2). Note that if an
axisymmetric flow has also a non-zero azimuthal component, the latter can be treated as a scalar
function. Then the axisymmetric vector basis should include only the radial and axial
components. An example of such basis, successfully used in several studies (see below) is
(,)=
=
, (3.4.3)
where
=
+(+ 1)
. As above, the zero divergence of (,) follows
from Eq. (A2), and the coefficients and are obtained by substitution of (,) in the
boundary conditions. Note also that the r-component of velocity vanishes at the polar axis = 0,
so that for flow in a full cylinder only three conditions in the radial direction must be additionally
satisfied. If the domain is a cylindrical layer (e.g., Taylor-Couette flow) then the polar axis is cut
out and one remains with the four boundary conditions, as in the Cartesian case.
Now consider Fourier modes of (3.4.1) corresponding to 0. Using the same idea as
in Eq. (3.1.1) we observe that
=
=
()
0+
0
. (3.4.4)
Obviously, the sum of two azimuthal components of this relation satisfies the boundary
conditions for . It is not clear, however, whether each of them satisfies the boundary
17
conditions separately. The author is not sure that this can be proved for a general case, but it can
be easily examined for no-slip conditions at = and = 0, . Since (=)= 0 the
derivative
also vanishes at =. Since the sum of two azimuthal components vanishes
at =, the azimuthal component of the first vector of the r.h.s also vanishes there, so that each
component satisfies boundary conditions in the radial direction. Similarly, we consider
()
at = 0, and conclude that it vanishes there because (, 0)=(,)= 0.
Then also the second azimuthal component vanishes at = 0, , and all the no-slip boundary
conditions for the azimuthal velocity, as well as the axis condition, are satisfied by each r.h.s.
vector of (3.4.4) separately. Thus, considering flows with no-slip cylindrical boundaries, we can
decompose into two independent bases, so that the whole flow will be represented as
= ()
(,)
+
()(,)+
()(,)
()
(3.4.5)
Here the functions (,) represent the axisymmetric part of the flow and are defined in
(3.4.3).The functions (,) and (,) represent the two vectors in r.h.s. of (3.4.4) and are
defined as
(,)=
0
, (3.4.5)
(,)=
0
, (3.4.6)
Here = 0 for ||= 1 and = 1 for ||> 1,
()=(+ 1)()+ 2().
Again, the zero divergence of the functions (,)() and (,)() follow
from Eqs. (3.4.2) and (A2), and the coefficients , , , and are defined after
18
substitution of (3.4.5) and (3.4.6) in the boundary conditions. The integer parameter q appears
because of different boundary conditions posed for ||= 1 and ||1 at the polar axis
(Canuto, et al., 2006; Gelfgat et al. 1999), which are
At = 0: = 0, = 0,
= 0
±0, ±0, ±= 0 (3.4.7)
||= 0, ||= 0, ||= 0
For no-slip conditions at the top, bottom and sidewall of the cylinder the coefficients , ,
, , , and are defined as
=
, =
(),=
, = 0 (3.4.8)
=()
, =
,=
(), = 0 (3.4.9)
=1, = 0, = 0, = 0 (3.4.10)
==== 0, ==
,==
; (3.4.11)
==
()()
()(), ==()()
() , > 0 (3.4.12)
=== 0, =1 (3.4.13)
This example of divergence-free basis functions built for cylindrical geometries also
shows how construction of a divergence free basis satisfying all the boundary conditions can be
approached in other orthogonal coordinate systems. The process can be noticeably simplified if
two periodic spatial coordinates are involved in the formulation of the problem. In these cases
only a one-dimensional basis will be needed. Alternatively, the divergence-free basis in
cylindrical and spherical coordinates with two periodic directions can be defined as in Dumas &
Leonard (1994) or Meseguer & Melibovsky (2007).
4. Inhomogeneous boundary conditions
If the problem has inhomogeneous boundary conditions, they can be included as
algebraic constraints. A better method is a change of variables so that all inhomogeneities are
moved from the boundary conditions to the equations. Then the boundary conditions become
homogeneous, and the corresponding basis functions can be built as in Section 3.4. We assume
19
that all the boundary conditions are linear. Such change of variables can use analytic, as well as
numerically calculated functions. Several examples of this are briefly described below.
The simplest example for the change of variables is convection in a box, which has
constant temperatures at the opposite sides, while all the other boundaries are perfectly thermally
conducting or perfectly insulated. In the first case the temperature varies linearly along these
boundaries, while in the second case temperature derivative normal o the boundary must vanish.
These boundary conditions are satisfied by the linear temperature profile, which corresponds to
the temperature distribution in a purely conducting case. For example, if for the dimensionless
temperature (,,), the boundary conditions in the x-direction are (= 0, ,)= 1,
(= 1, ,)= 0, the change of variable is =(1)+ (,,). The function (1)
satisfies all homogeneous and inhomogeneous boundary conditions, so that all the boundary
conditions for new unknown function (,,) are homogeneous. This change of variables was
applied in all cited works of Gelfgat that treated convection in rectangular cavities starting from
Gelfgat & Tanasawa (1994).
The inhomogeneities can be excluded from the boundary conditions by extracting a
known analytical function from the solution only if the boundary conditions are continuous,
including continuity at the corners of the computational domain. Another example of this is
parabolic heating of a vertical cylinder that was considered in Gelfgat et al. (2000, 2001b). The
dimensionless temperature at the cylindrical sidewall was prescribed as (= 1, )=()=
(1
), 0 , and was zero at the top and bottom, = 0, . Since the function ()
vanishes at the top and the bottom, the simplest change of variables in this case is =()+
(,), which was applied in the mentioned studies.
Clearly, when the boundary conditions are discontinuous, use of a simple analytic
function for the change of variables becomes problematic. Such a function can be built, for
example, as a solution of Laplace equation with discontinuous boundary conditions. This
analytic solution will suffer from Gibbs phenomenon which may destroy the convergence of the
whole process. On the other hand, a low-order numerical solution can smooth the discontinuity
up to an acceptable level, as in many calculations of lid-driven cavity flow, however this will
take us too far from our Chebyshev-polynomial-based Galerkin approach. Thus, for calculation
of the lid-driven cavity flow in Gelfgat (2005) we used analytically smoothed boundary
condition, then solved the Stokes problem with the smoothed boundary conditions, and then used
20
the Stokes problem solution for a change of variables. The Stokes problem was solved using the
same Galerkin approach.
In the studies of swirling flows in a cylinder with rotating lid, as well as independently
rotating top, bottom and sidewall of the cylinder (Gelfgat et al., 1996a,b, 2001; Marques et al.,
2003) we solved the Stokes problem for the azimuthal velocity component. A similar Galerkin
method in scalar formulation was applied. The boundary conditions with discontinuities in the
corners were included as additional algebraic constraints by adding collocation points along the
boundaries.
A more complicated case was treated in
Erenburg et al. (2003). There we considered
convection in a rectangular cavity with partially
heated and partially insulated sidewall as
sketched in Fig. 1. All the boundaries are no-
slip. The dimensionless boundary conditions for
the temperature are (here = ,
=
,=
)
(;= 0, )= 0, (4.1)
(= 0,1; )= 1, (4.2)
(= 0,1; < >)= 0. (4.3)
To arrive at a formulation with continuous and
homogeneous boundary conditions on all the
boundaries for a single unknown function, we
decompose the temperature into two functions
(,,)=(,,)+(,,) (4.4)
where (,,) is the new unknown function for which a continuous set of boundary conditions
is required, i.e., (;= 0, )= 0,
(= 0,1; )= 0 (4.5)
x
0
H
y
h
1
h
2
θ=θ
cold
0
=
∂
∂
r
θ
0=
∂
∂
r
θ
0=
∂
∂
r
θ
0=
∂
∂
r
θ
θ=θ
cold
θ=θ
hot
θ=θ
hot
L
Fig. 1. Sketch of the temperature boundary conditions
of the problem of Erenburg et al. (2003).
21
The function (,,) is used to adjust the boundary conditions for (,,) to (4.1)-4.3).
Therefore, the boundary conditions for (,,) are
(;= 0, )= 0, (4.6)
(= 0,1; )= 1 , (4.7)
(= 0,1; < >)= 0 (4.8)
To avoid the appearance of an additional source term in the energy equation we also require that
(,,) be a solution of the Laplace equation,
= 0. (4.9)
The solution of problem (4.6)-(4.9) can be represented as
(,,)=(,,)+(,,) (4.10)
where (,,) is the part of the solution of (4.6)-(4.9) corresponding to = 0 and (,,)
is the part dependent on . Both functions and are calculated numerically by the global
Galerkin method inside the cavity and collocation points at the sidewalls. Obviously, the part
(,,) is defined by the geometry of the problem only, and is time-independent, so that it
must be calculated only once. The problem formulation for (,,) is straight-forward, and its
solution can be presented as =, where is the operator defining the problem, and
approximated by its Galerkin/collocation projection. The representation of the temperature (4.4)
now becomes (,,)=(,,)+(,,)=+(+) , (4.11)
where is the identity operator. The energy equation becomes
(+)
+()(+)=
(+)() . (4.12)
Thus, after the function and the operator (+) are calculated, the remaining problem for
is defined with the homogeneous boundary conditions (4.5) only. Other details can be found in
Erenburg et al. (2003).
5. Basis functions for two-fluid flow and boundary conditions at liquid-liquid interface
Here we give an example of basis functions that were used to calculate a two-phase flow
with a capillary interface separating two liquids. A two-fluid Dean flow sketched in Fig. 2 was
considered in Gelfgat et al. (2001d). The two fluids occupy adjacent thin cylindrical layers,
+ and ++, respectively, with the assumption =
1. The
22
two fluids are separated by the border =+. The base flow in both fluids is driven by a
constant azimuthal pressure gradient. Note that while the pressure is then a multi-valued function
of ,
is not and can be considered as an imposed bulk force This formulation is an
extension of the classical Dean (1928) problem to two-fluid system and is a simplified model of
flow in a curved channel. Here we leave all the details concerning the evaluation of the base flow
and the formulation of the stability problem to Gelfgat et al. (2001d), and focus only on the
boundary conditions and incorporation of them into the basis functions. The three-dimensional
velocity perturbation in cylindrical coordinates is defined as =(),(),()(+
+). For convenience, we define a new dimensionless coordinate =()
, and
define =()
. The dimensionless amplitude of the interface perturbation is . Indices 1
and 2 denote the variables in each sublayer, =
and =
is the ratio of
densities and viscosities, respectively. After the axial velocity and the pressure are
eliminated, the conditions for the radial and azimuthal components at the bounding surfaces and
the interface read = 0: ==
= 0 (5.1)
= 1: ==
= 0 (5.2)
=: =,= (5.3)
=
(5.4)
=
(5.5)
+=
+ (5.6)
=1
(5.7)
12 2
1
=212 12+
+
2
()
()
(5.8)
To construct basis functions, we start from non-deformable interface. In this case we add
== 0 to the boundary condition (5.3) and omit (5.7) and (5.8). Then the unknowns ()
and () are approximated by truncated series
Fig. 2. Sketch of the two-fluid Dean flow problem
1
2
b
a
r
θ
d
V
θ
23
()=()
(), ()=()
() , (5.9)
where
()=
()
, 0
()
,1
, (5.10)
()=
()
, 0
()
,1
. (5.11)
Here the superscripts (1) and (2) denote the sublayers. The coefficients
(),
(),
(), and
()
are obtained after substitution of the basis functions (5.11) into the boundary conditions (5.1)-
(5.6). The inner product is defined as
,=()
()=()
()+()
(), (5.12)
and after application of the Galerkin method, the time-dependent coefficients () and ()
are the same for the whole domain.
For the linear stability problem with deformable interface the solution is represented as
()=()()+()
(), ()=()
() . (5.13)
The bases () and () remain unchanged. An additional function () is introduced to
satisfy the boundary conditions for a deformable interface. Its choice is arbitrary. In our case we
define it as
()=
()
, 0
()
,1
. (5.14)
The coefficients
() and
() are used to satisfy the boundary conditions (5.1), (5.2), (5.4) and
(5.6) subject to the normalization condition = 1. Choice of the index is arbitrary, e.g.,
= 0.
With the normalization condition = 1 applied, the coefficient can be interpreted
as the amplitude of the radial velocity at the deformed interface. This coefficient, and the
24
interface amplitude , are defined by the two remaining boundary conditions (5.7) and (5.8).
Thus, the Galerkin projections of the governing equations together with the boundary conditions
(5.7) and (5.8) form a closed algebraic system for calculation of the coefficients and
together with two additional unknown scalars and . The stability problem reduces to a
generalized eigenvalue problem. In Gelfgat et al. (2001d) coefficients of the basis functions
(),
(),
(),
(),
(), and
() were obtained by means of computer algebra.
6. The dynamical ODEs system for time-dependent coefficients
6.1. General expressions to be used coding the calculations
In the following we assume that all the inner products are defined with unit weight. We
also assume that all the necessary changes of variables are made, so that the boundary conditions
of all the unknown vector and scalar fields are linear and homogeneous. Then, after the basis
functions are constructed, and the Galerkin projections are made, and the pressure is eliminated
by Eq. (2.20), we arrive at an ODE system (2.21) for the time-dependent coefficients. We store
all the unknown time-dependent coefficients of the problem in the vector ()={()}
,
where is the total number of scalar unknowns (degrees of freedom). Note that the vector ()
contains velocity coefficients, as well as coefficients of all the other possible unknowns, e.g.,
temperature and/or concentration, excluding pressure. Then the resulting dynamic ODE system
has the following form (the Einstein summation rule is assumed)
=++ . (6.1.1)
Here is the Gram matrix, , , and are projections of the linear, nonlinear and bulk
force of the momentum equation (2.7), respectively. The transport equation for temperature,
concentration, electric and magnetic fields, etc., arrive to similar ODE systems that can be
complicated by non-linear terms not belonging to the material derivative. This dynamical system
has several nice properties that follow from the fact that the basis functions satisfy all boundary
conditions, and are divergence-free. From Green’s theorems for a scalar function and for a
divergence-free velocity
,=,, ,=×,× . (6.1.2)
25
It follows that the matrices corresponding to the dissipative terms are symmetric and negative
definite independently on the truncation number. Furthermore, from the conservation properties
(),= 0, (),= 0 (6.1.3)
it follows that = 0 (6.1.4)
for any truncation number. This means that the non-linear term always conserves the momentum.
Additionally, these properties yield a very convenient tool for code debugging.
Computation of steady state flows reduces to an algebraic system of quadratic equations
++= 0 , (6.1.5)
for which we do not need to consider the Gram matrix. The application of Newton iteration is
straightforward and requires computation and inversion of the Jacobian matrix
=++ . (6.1.6)
Linear stability analysis of the calculated steady states reduces to computation of the eigenvalues
of another Jacobian matrix, which includes the inverted Gram matrix
=,
=
++=+
+
. (6.1.7)
The inverse of the Gram matrix is also needed for straightforward time integration, for which the
dynamical system (6.1) has the form
=
++=+
+ , (6.1.8)
where matrices multiplied by are denoted by .
Explicit representation of the dynamical system (6.1.8) allows one to perform additional
analytical evaluations required by weakly non-linear analysis of bifurcations. This was
implemented for the Hopf bifurcation in Gelfgat et al. (1996) and Gelfgat (2004). Assume that
with increasing Reynolds number, at =, a complex conjugate pair = ± of leading
eigenvalues of the problem (6.1.7) crosses the imaginary axis. Then the instability sets in as a
Hopf bifurcation if all the conditions of the Hopf theorem hold, which is the most common case.
We are interested in an asymptotic approximation of the oscillation period and amplitude at
small super-criticality. Assume that is the steady state at the critical point, and and are
the left and right eigenvectors corresponding to the eigenvalue =. We also denote the
r.h.s. of the dynamic system (6.8) as (;)=. Then, according to Hassard et al. (1981), the
oscillating state, i.e., the limit cycle, is approximated as
26
=++() (6.1.9)
()=
[1 + +()] (6.1.10)
(,)=()+
+ () (6.1.11)
Here is a formal positive parameter, is the super-criticality, is the oscillation
period, and (,)is the asymptotic oscillatory solution of the ODE system (6.1.8) for the
Reynolds number defined in (6.1.9). This asymptotic expansion is defined by two parameters
and , which are calculated using the following procedu (Hassard et al., 1981)
=()
, =
[()+], =+=
, (6.1.12)
The parameter is obtained as
=
+
2||
|| (6.1.13)
= 2, = 2, = 2. (6.1.14)
The vectors and are the second derivatives of the r.h.s., and is the third derivative in
the complex plane , :
=
[+();], =
[+();]
(6.1.15)
=
2+++; , (6.1.16)
and the vectors and are solutions of the following systems of linear algebraic equations
( is the identity matrix and denotes the Kronecker product)
=,
2= , =() (6.1.17)
The eigenvalue derivative (6.1.12) can be easily evaluated numerically. However, the most
difficult part of this calculation is evaluation of the second and the third derivatives of the right
hand side of the ODE system (6.1.15) and (6.1.16). These derivatives must be evaluated in the
complex plane. The number of degrees of freedom is large, so that accurate enough
differentiation by finite differences that will involve small increments of can be problematic.
However, the explicit form of (6.1.8) allows for analytical calculation of the derivatives. The
result is ,=
()()()()+()()()() , (6.1.18)
27
,=
()()+()() , (6.1.19)
=
+2,+ 2, , (6.1.20)
where superscripts () and () denote real and imaginary parts, respectively. These analytical
expressions allow one to compute the asymptotic expansions (6.1.9)-(6.1.11) without significant
loss of accuracy. The CPU-time requirements for such calculations are comparable with the
calculation of two steady state solutions and their spectra. The sign of determines whether the
bifurcation is sub- or super-critical.
6.2. Main computational difficulties
All the computations described in the previous section, are restricted by two main
difficulties. To describe these, we note that in all the studies where this method was successfully
applied, the number of basis functions in one direction was between 20 and 70. Therefore, to
make some estimates, we will address three types of truncation in one direction with 30, 60 and
100 basis functions for both two- and three-dimensional problems.
The first difficulty is connected with the Gram matrix . In two-dimensional and quasi
two-dimensional cases, e.g., 3D instability of an axisymmetric base flow, there is no problem
storing the Gram matrix in memory, nor in computing its inverse. In fact, even with the
truncation number 100, the Gram matrix consists of blocks of order 1002=104, which leads to an
order of 108 non-zero entries, which fits in several Gb memory. This matrix, which is symmetric
and positive definite, can be inverted by Choleski decomposition, which is much faster than
Gaussian elimination. This inverse must be computed only once for each specific problem and
can be stored on disk. However, for all the calculations, except the Newton iterations, the r.h.s.
of dynamic system must be multiplied by . If the truncation number is small, the matrices
= and
= in the dynamical system (6.1.7) can be computed and stored before
other heavy computations begin. At larger truncation number the storage of matrix
becomes
impossible (see below), which makes the time-dependent calculations too slow. At the same
time, the stability analysis, as well as the weakly non-linear analysis, require only a few, usually
less than 10, evaluations of the Jacobian matrix
, thus making the computational process
affordable.
28
Treatment of the Gram matrix in a three-dimensional formulation is much more difficult.
Here the largest block to be inverted has order of ((2)) elements, where M is the truncation
number. Storage of such large matrices becomes problematic already at truncation numbers
30, and is unaffordable at =100. Thus, among all the possibilities described, only
steady state calculations are possible. A solution for this can be orthonormalization of the set of
the basis functions, which is discussed below.
The second difficulty is the numerical evaluation of non-linear term in (6.1.5) and (6.1.7),
which requires the order of ((2)) multiplications, assuming that the matrix
is stored and
evaluation of its terms does not require additional operations. Again, it can be affordable for the
steady state, stability, and weakly non-linear calculations when 2D and quasi-2D problems are
solved, because all these require very few evaluations of the r.h.s. In the fully 3D cases it seems
to be not feasible, unless some additional evaluations of the non-linear terms are performed.
6.3. Treatment of non-linear terms
As a simplest example of handling the non-linear terms and avoiding too many
multiplications, we consider the Burgers equation in the interval 01
+
=
, (0, )=(1, )= 0 . (6.3.1)
The initial condition used in Gelfgat (2006) was (, 0)=(2)+()2
. We seek the
solution as a truncated series
(,)=()()
, ()=()(), (6.3.2)
where the basis functions () are constructed as linear superpositions of the Chebyshev
polynomials to satisfy the boundary conditions (see Eqs. (A3)). After the Galerkin projections
are applied we obtain the ODE system (6.1.1), where coefficients () are stored in the vector
. For this problem the matrices in (6.1.1) are defined as
=,, =,, = 0, =
,, (6.3.3)
and evaluation of the non-linear term requires operations.
To reduce the number of multiplications, we notice that
is a polynomial of order
++ 1 that satisfies the boundary conditions of the problem, so that we can express it as a
series of (): ()
()=
() (6.3.4)
29
The coefficients can be evaluated analytically using Eqs. (A7) and (A12) in the following
way ()
()=()()[()
()] = (6.3.5)
=()()4 ()
[()
]
4(+ 2) ()
[()
]
= 2 ()+()()+()
[()
]
2(+ 2) ()+()()+()
[()
]
= 2 ()()+()()
[()
]
2(+ 2) ()()+()()
[()
]
= 2 ()+()
[()
]
2(+ 2) ()+()
[()
]
=4 ()+()
[()
]
2(+ 2) ()+()
[()
]
[()
]
Defining additionally = 0, noticing that [(1)2
]+ 1 = [(+ 1)2
], and comparing
the above result with (6.3.4) we observe that the coefficients can be assembled by the
following procedure. Starting from = 0,
[()
]=[()
]+ 2(+ 2)[()
], (6.3.6)
12[(+ 1)2
]20
30
[()
]=[()
]2(+ 2)[()
],
12[(+ 1)2
]+0
For = 0 to =[(1)2
]:
=+ 4, 1220
=4, 12+0
Now, using (6.3.4), we form a new set of time-dependent coefficients:
()()
, ()
()=()
()
(), (6.3.7)
()=()()+
, =+ (6.3.8)
And finally,
()()
, =
,()()
, =()
()
, (6.3.9)
Now, we can estimate the number of multiplications required. The coefficients , and the sums
in (6.3.8) depend only on the basis functions and, therefore, can be computed only once in the
beginning of the computational process. Computation of the coefficients () requires 2
multiplications and is the most CPU-time consuming part. Then, evaluation of (6.3.9) requires
2(1) multiplications providing that all the inner products are pre-computed. Since the
operations in (6.3.8) and (6.3.9) are easily scalable, vectorization and/or parallel computing can
additionally speed up the calculations.
Returning to the non-linear terms of momentum equation, we observe that in the case of
no-slip conditions the non-linear terms
,
,
, etc., satisfy the no-slip
boundary conditions for ,, and , respectively. Thus, these terms can also be decomposed into
series of appropriate basis functions, which will lead to a similar decrease in the number of
required multiplications. When boundary conditions are more complicated, e.g., a stress-free
boundary, one can add additional functions in which only boundary conditions satisfied by the
non-linear terms are implemented. Alternatively, regardless of boundary conditions, the non-
linear terms can be represented as Chebyshev polynomial series, which will also decrease the
number of multiplications.
6.4. Orthogonalization and other polynomial bases
31
In this section we address two questions: does (i) orthogonalization of the basis or (ii)
another polynomial basis change the final result? The answer is “no”, but it requires some
additional explanations.
After choosing the truncation number we seek the solution in the form of (3.3.8). The
solution belongs to the linear space =
(,),
(,),
(,) as in (3.3.8). This space
consists of divergence-free vectors that satisfy all the LHBC of the problem, and their
components are polynomials of maximum order (,),(,),(,)+ 4 in the x-direction,
with similar expressions in the two other directions. We denote the order of space as and
store all the basis functions of (3.3.8) in a set of vectors ={}
. Clearly, this set forms a
basis in , ={}. Assume now another basis in , denoted as
={
}
. The
connection between the two bases is given by a matrix of order as
=, =
. (6.4.1)
Elements of the matrix are solutions of the following system of linear algebraic equations
(summation over repeating indices is assumed)
,
=,
(6.4.2)
Equation (6.4.1) can be interpreted as a transformation to another polynomial basis, a particular
case of which is an orthonormal polynomial basis. In this case the matrix is then the operator
of the Gram-Schmidt or Householder orthogonalization procedure. Assume now that projection
of the solution on each of the bases is described by coefficients and . Since these
coefficients describe the orthogonal projection of the vector onto the same space, the result of
projection onto either basis must be identical, i.e.,
=
= , (6.4.3)
from which it follows that
=, =
,
=()=() (6.4.4)
The Galerkin procedure applied with either of the two bases will result in two different ODE
systems similar to (6.1.1):
=++ , =+
+ , (6.4.5)
where =,, =
,, =,,=, (6.4.6)
32
=
,
, =
,
,
=
,
,=,
(6.4.7)
Note that the two ODE systems in (6.4.5) describe the orthogonal projection of the residual on
the same linear space , so that the result again must be identical. However, it is not clear yet
whether the coefficients and yielded by solution of the two systems will be connected via
Eq. (6.4.4). Let us evaluate how the matrices in (6.4.6) and (6.4.7) are connected.
=,
=,= , (6.4.8)
=
,
=,= , (6.4.9)
=
,
=
,= , (6.4.10)
=
,
=,= . (6.4.11)
Consider now how all the terms of the second system of (6.4.5) are expressed via matrices and
unknowns of the first system =
1= , (6.4.12)
=
1= , (6.4.13)
=
1
1= . (6.4.14)
Substituting (6.4.8), (6.4.12)-(6.4.14) into the second system of (6.4.5) we obtain
=++ , (6.4.15)
and multiplying (6.4.15) by we return to the first system of (6.4.5). This proves that both
systems of (6.4.5) yield identical solutions if their initial conditions are connected via eq. (6.4.4).
To conclude, seeking other polynomial basis functions is useless, since we’ll arrive to
exactly the same approximate solution. On the other hand, the orthonormalization procedure can
be meaningful, since it does not change the solution, but avoids inverting the Gram matrix.
7. Inner products with arbitrary weight
For a scalar problem, e.g. Orr-Sommerfeld or Burgers equations, the choice of the weight
function in the inner product (2.18) is arbitrary. In the case of unit or Chebyshev weight, the
inner products can be evaluated analytically using properties of the Chebyshev polynomials
listed in Appendix A. In other cases the Gauss quadrature formulae can be efficiently applied.
33
An appropriate choice of the weight function can drastically improve the convergence, as was
demonstrated in Gelfgat (2006) for the Burgers equation.
For the incompressible Navier-Stokes equation, use of the unit weight function allows
one to eliminate the pressure by the Galerkin projection. The unity weight also yields important
conservation properties of the resulting ODE system (6.1.3) and (6.1.4). At the same time, if the
weight function can be optimized such that the convergence is noticeably improved, then the
total number of degrees of freedom in the resulting dynamical system can be decreased. In this
case it can be reasonable to give up the nice properties of the unit weight and proceed with the
optimized one. Then it will be necessary to solve the pressure equation (2.11), so that the ODE
system with the algebraic constraints (2.16), (2.17) will be considered. In the following we
follow Gelfgat (2006) to show how the algebraic constraints can be removed in the framework of
the Galerkin approach.
First of all, the Laplacian operator in pressure equation (2.11) must be supplied by
boundary conditions. It was shown in Gelfgat (2006) that the boundary conditions proposed by
Gresho & Sani (1987), which are limits of the momentum equation at the boundaries, yield a
correct pressure field. The boundary conditions on the boundary are
=
()
+ , (7.1)
Where is the normal to , and the boundary conditions for velocity are assumed to be steady.
Note that Rempfer (2006) argued that the numerical solution of the pressure problem (2.11),
(7.1), together with the momentum equation (2.7), do not yield a divergence-free solution for
velocity. The global Galerkin method with divergence-free velocity basis functions described
here does not have this problem, because the continuity equation is satisfied analytically by the
basis functions, before the numerical process starts. Thus, any approximation of a solution is
analytically divergence free.
For the following we represent the pressure as a truncated Chebyshev series
(,,,)=()()()()
,, . (7.2)
Here we cannot introduce any boundary conditions into the basis functions, because we cannot
propose any sufficiently general change of variables that will make the boundary conditions (7.1)
homogeneous for the pressure, so that they will be incorporated in the pressure basis functions.
To obtain a problem for the unknown coefficients () we perform Galerkin projections of the
34
residuals of (2.11) and (7.1) on the Chebyshev basis (7.2), and require that the projections
vanish. In other words, we apply the Galerkin method separately in the domain and on the
boundaries. Clearly the total number of unknowns () must be equal to the total number of
equations used. Recalling that the velocity coefficients are stored in the vector , we store the
additional coefficients of pressure () in the vector . After the Galerkin process is
completed, the system of linear algebraic equations for has the following form
()=
()+
()+
()+() (7.3)
These equations are formed from the projections of Eqs. (2.11) and (7.1). The superscript () is
used to underline that all the matrices belong to the pressure problem. The matrix
() is singular
because the pressure is defined to within an additive constant. This singularity can be easily
removed by, e.g., an additional requirement = 0, after which the matrix () is regular and
its inverse is denoted as . The Galerkin coefficients of velocity must be calculated using the
equations (2.16), which take the following form
=+++ . (7.4)
Here the first term of the r.h.s. is the projection of the pressure gradient onto the velocity basis.
Substituting from Eqs. (7.3) into Eqs. (7.4) we obtain
=+
+ , (7.5)
where
=(), =+(),
=+(), =+() (7.6)
Thus, after some analytical and numerical evaluations we obtain the ODE system (7.5), whose
structure is equivalent to that of (6.1.1). Generally, the matrices of (7.5) do not obey the
properties (6.1.3) and (6.1.4), however everything else that we have written about the system
(6.1.1) is applicable also to (7.5).
Some numerical experiments comparing convergence of the Galerkin method with
Chebyshev and unit weights are reported in Gelfgat (2006). There the lid-driven cavity flow and
convection in a laterally cavity were taken as test problems. It was found that the Chebyshev
weight allows for a better resolution of boundary layers in the convection flow, but slows down
the convergence for the lid-driven flow. This was attributed to the problem of approximating the
corner discontinuities.
35
Fig. 3. Comparison of numerical results with the experimental
phograph of Escudier (1984). From Gelfgat et al. (1996). The flow at
H/R=3.25, Re=2752. (a) calculation with the Galerkin method using
34×34 basis functions. (b), (c) calculation with the finite volume
method using 100×100 and 200×200 grids, respectively.
The optimization of the weight function has never been considered for a realistic fluid
dynamics problem. The only optimization example is given in Gelfgat (2006) for the Burgers
equation. Considering weight functions of the form of () , 0, it was found that the
convergence is fastest at = 1.3.
8. Solved problems and other applications of the method
The effectiveness of the Galerkin approach follows from a decrease in the number of
degrees of freedom of a numerical model. The decrease is usually of an order of magnitude or
more. One of the earliest
examples of this is shown in Fig.
3. There we consider flow in a
cylinder with rotating lid. Steady
states of this flow exhibit the
vortex breakdown phenomenon,
which was experimentally
studied by Escudier (1984). At
certain Reynolds number a weak
reverse circulation attached to
the cylinder axis appears. In tall
cylinders up to three distinct
recirculation zones are observed.
The intensity of the reverse
vortices is 3-5 orders of
magnitude less than that of the main meridional circulation, which makes its numerical modeling
quite challenging. We show in Gelfgat et al. 1996 that the size and position of the reverse
circulations is well reproduced with 34 with 34×34 basis functions (Fig. 3a), as well as with
200×200 finite volume grid (Fig. 3c), but is resolved inaccurately with the 100×100 grid (Fig.
3b). The total number of degrees of freedom of the Galerkin method is the number of unknown
Galerkin coefficients of the meridional velocity vector and that of the azimuthal velocity
component, which separates as a scalar in the axisymmetric formulation. In the finite volume
approach this is the number of unknown functions multiplied by the number of grid nodes.
36
Clearly the total number of degrees of freedom consumed by the Galerkin method, 234=
2312, is smaller than that of the finite volume method, 3200=120,000, by about 1.5 orders
of magnitude.
The next example presented in Fig. 4 is a thermocapillary convective flow in a laterally
heated cavity with the aspect ratio length/height=4. The velocity boundary condition at the upper
surface is = 0,
=
, (8.1)
where is the dimensionless temperature, =
, and and are the Marangoni and
Prandtl numbers, respectively (other details are in Gelfgat, 2007a). The problem was treated with
the truncation of 50×20
basis functions with 100
collocation points at the
upper surface to satisfy
the boundary condition
(8.1) The critical
Marangoni number,
corresponding to
transition from steady to
oscillatory flow regime,
was calculated to be
=4781 and the
dimensionless critical
frequency (imaginary part of the leading eigenvalue) =6674. This result was never
published because the value of seemed to be too high compared with the values already
known for buoyancy convection (Gelfgat et al., 1999a). Owing to the computer restrictions of
that time, the convergence could not be rigorously checked. Later, the same problem was solved
using a 800×200 stretched finite volume grid (Gelfgat, 2007a) and the result was =4779
with the same value of the critical frequency. The convergence of the critical values with
resolution obtained by the finite volume method was found to be very slow. The reason for that
is seen in Fig. 4. The streamlines and the isotherms are smooth, so that the velocity and
temperature fields can be easily calculated. At the same time, the perturbation patterns exhibit
Fig. 4. Patterns of flow and amplitude of the most unstable perturbation at the
critical Marangoni number for thermocapillary convection of low-Prandtl-
number fluid (Pr=0.015) in a cavity of aspect ratio length/height=4. From
Gelfgat (2007a).
37
very steep maxima, which must be numerically resolved to obtain correct critical values. We
observe here again that the Galerkin method yielded the correct result with a much smaller
number of degrees of freedom.
The smaller
number of degrees of
freedom, as well as
analytic representation of
the Jacobian matrix
(6.1.6), allows one to
effectively apply Newton
iteration for calculation of
the steady states. To
follow different solution
branches one also can
apply arc-length or similar
continuation technique.
This is illustrated in Fig. 5
for the convection in a
cavity with partially heated sidewall (Erenburg et al., 2003), the boundary conditions for which
were discussed in section 4. Since the boundary conditions are symmetric, the flow at low
Grashof number is symmetric. The symmetry is broken at =180. The diagram in Fig. 5
shows difference between the Nusselt numbers calculated at the left and right vertical
boundaries, so that in the symmetric state the difference is zero. After the symmetry breaks, we
observe several interconnected solution branches with qualitatively different flow patterns.
Those depicted in red are stable, and those depicted in color are oscillatory unstable. Although
most of the steady state branches are unstable, we speculate that there exist multiple oscillatory
states with similar flow patterns.
Fig. 5. Bifurcation diagram for convection in a vertical cavity with partially
heated sidewall. Red lines correspond to stable steady states, blue lines to
unstable ones. Pr=10. From Erenburg et al. (2003).
38
The most well-known results obtained with this method are stability diagrams of swirling
rotating disk – cylinder flow and buoyancy convection flows in laterally heated rectangular
cavities. The first
results on the three-
dimensional instability
of rotating disk –
cylinder flow were
obtained in Gelfgat et
al. (2001a) using the
Galerkin method with
30×30 basis functions.
Since then several
research groups
validated these results
experimentally and
numerically. These
comparisons are very
convincing and are shown in Figs. 6 and 7. According to our results, the instability is
axisymmetric for the cylinder aspect ratio (height/radius) varying between 1.6 and 2.7, and is
three-dimensional outside of this interval. The three-dimensionality sets in with the azimuthal
wavenumber = 2 at small aspect ratios, and with = 3 or 4 in taller cylinders. Several later
studies tried to reproduce these results either by straightforward integration in time, or by means
of stability analysis, and fully confirmed our conclusions. The quantitative comparison was
carried out for the critical Reynolds numbers and critical frequencies, as well as for the azimuthal
mode number. It was possible also to confirm values of the aspect ratio at which the modes
replace each other.
Fig. 6. Stability diagram for flow in a cylinder covered by rotating disk. The
lines correspond to results obtained by the Galerkin method (Gelfgat et al.,
2001) for different azimuthal wavenumbers k in (3.4.1). Symbols show
results independent numerical studies.
39
A similar comparison, but
experimental, was carried
out by Sørensen et al (2006,
2009). Their result is shown
in Fig. 7. The oscillations
were measured by LDA,
and the flow azimuthal
periodicity by PIV. The
symbols in Fig. 7 show
experimentally measured
points, and their color
corresponds to the
azimuthal wavenumber as
shown in the figure. The
pioneer results of Escudier
(1984) are also shown. Taking into account experimental uncertainties, the agreement between
experiment of Sørensen et al (2006) and the numerical predictions of the Galerkin method is
quite impressive. The
agreement with earlier
results of Escudier
(1984) is observed only
for the axisymmetric
mode, but his
experiments were based
only on visualizations by
the aluminum powder,
and the instability was
observed only as
oscillations of the vortex
breakdown bubbles.
Other stability
Fig. 7. Stability diagram for flow in a cylinder covered by rotating disk. The
lines correspond to results obtained by the Galerkin method (Gelfgat et al.,
2001) for different azimuthal wavenumbers k in (3.4.1). Symbols show
experimental results.
Fig. 8. Stability diagram for convective flow in laterally heated cavities. The
curves color corresponds to the number of convective rolls in the flow
pattern. The flows are stable below or between the curves of the same color.
Dashed regions correspond to the stability regions of similar flows with
broken rotational symmetry. Pr=0. From Gelfgat et al. (1999)
0.E+00
2.E+05
4.E+05
6.E+05
8.E+05
1.E+06
2 3 4 5 6 7 8 9 10
A
Gr
cr
40
results obtained by the described Galerkin method for similar swirling flows can be found in
Gelfgat at al. (1996b) for flow in a cylinder with independently rotating top and bottom, and in
Marques et al. (2003) for independently rotating top and sidewall.
The neutral curves shown in Fig. 6 are plotted through up to a hundred calculated critical
points. The next example, relating to the oscillatory instability of buoyancy convection flows in
laterally heated cavities and shown in Fig. 8,
required several hundreds of critical points to
complete the study. The calculations were
performed with up to 60×20 basis functions. With
increasing cavity aspect ratio A= length/height, and
at large enough Grashof number, the single
convective cell splits into several cells. The
number of cells grows with the aspect ratio. At the
same time several steady states with a different
number of rolls can be stable at the same set of
governing parameters, as shown in Fig. 9. The
transition from one number of cells to another takes place at points where a neutral curve of a
certain color continues with a different color. At these points the flows with, e.g., two and three
rolls are indistinguishable. Other results on stability of convection in rectangular cavities can be
found in Gelfgat et al. (1996, 1999a,b), Erenburg et al. (2003), and Gelfgat (2004). One
particularly interesting result reported in Gelfgat (2004) showed that weakly non-linear
approximation of limit cycles (6.9)-(6.11) yields results that are very close to those obtained by
independent straightforward time integration.
Several studies have been devoted to three-dimensional instabilities of axisymmetric
buoyancy convection in vertical cylindrical containers. These studies were started in Gelfgat et
al. (1999c), where we were able to reproduce a nice experimental result showing the breaking of
axisymmetry leading to a spoke pattern with azimuthal wavenumber = 9. Later we studied
cylinders with non-uniformly heated sidewalls that mimicked conditions of Bridgman crystal
growth (Gelfgat et al., 2000, 2001b). Later works were devoted to axisymmetric flows driven by
rotating or traveling magnetic field (Gelfgat & Gelfgat, 2004; Gelfgat, 2005). Most of these
results were reviewed in more detail in Gelfgat & Bar-Yoseph (2004).
branch 1
branch 2
branch 3
Fig. 9. Three distinct stable steady states
found at Pr=0, A=7, Gr=88,000. From Gelfgat
& Bar-Yoseph (2004).
41
Additional opportunities are provided by analytical representation of numerical solution
via the Galerkin series. Clearly, one can differentiate or integrate the series without any
noticeable loss of accuracy, contrary to low-order methods. An obvious example is the
calculation of flow trajectories using a previously calculated steady or time-dependent flow.
Since the velocity field is defined analytically in the whole domain, wherever the liquid particle
arrives, its velocity is known without the need to interpolate between grid nodes. This fact was
used in Gelfgat (2002), where trajectories were calculated over very long time to obtain a
Poincare map in the midplane.
Another application of the divergence-free bases (3.3.2), (3.3.3), and (3.3.6) is
visualization of three-dimensional incompressible flows, as described in Gelfgat (2014). Without
going into detail, we only mention that projection of flow on each divergence-free set can be
interpreted as a divergence-free projection on coordinate planes =,=,=
, which results in two-component divergence-free fields. These can be described by an
analog of the streamfunction. Assembling all the planes, e.g., =, we obtain a scalar 3D
function whose isosurfaces are tangent to the projected vectors. Three such projections of three
sets of coordinate planes complete the visualization of a three-dimensional flow field. Details
and illustrations can be found in Gelfgat (2014).
9. Similar approaches in studies of other authors
As mentioned, the idea to use linear superpositions of Chebyshev polynomials for
definition of basis functions satisfying linear and homogeneous boundary conditions was
introduced by Orszag (1971a,b) for the homogeneous two-point Dirichlet problem. Since then,
similar linear-superposition-based basis functions were used to solve one-dimensional problems
for, e.g., Orr-Sommerfeld and boundary layer equations, by Holte (1983), Pasquarelli (1991),
Yueh & Weng (1996), Yang (1997), Borget et al. (2001), Yahata (2001), Bistrian et al. (2009),
and Buffat & Le Penven (2011). In these works the basis functions were constructed from the
Chebyshev polynomials. Recently, Wan & Yu (2017) applied the same idea to Legendre
polynomials. Grants & Gerbeth (2001), Uhlmann & Nagata (2006), and Batina et al. (2009) used
the same approach for a two-dimensional flow field, but their basis functions were not
42
divergence-free. Picardo et al. used linear superpositions for a two-fluid problem, as was done in
Gelfgat et al. (2001d).
Moser et al. (1983) proposed to multiply linear superpositions of the Chebyshev
polynomials by powers of the Chebyshev weight function in order to make better use of the
polynomial orthogonality properties. For problems with two periodic directions, these authors
constructed a divergence free basis, in which the non-periodic direction was treated by linear
superpositions of the Chebyshev polynomials multiplied by additional weight-dependent
functions. Such functions were used for either coordinate or projection bases in the weighted
residuals method by Ganske et al. (1994), Goddeferd & Lollini (1999), Kerr (1996). It should be
noted that multiplication by either function makes evaluation of derivatives and computations of
their Galerkin projections more complicated.
Yahata (1999) solved a problem of buoyancy convection in laterally heated cavities
similar to those treated by Gelfgat & Tanasawa (1994) and later by Gelfgat et al. (1997 1999a,b).
He used linear superpositions of Chebyshev polynomials to construct basis functions for the
temperature and the streamfunction with subsequent orthogonalization of the basis. The inner
product was formulated with an arbitrary weight function, however it is not clear which weight
was used. By evaluating derivatives of the streamfunction basis of Yahata (1999) one would
obtain the two-dimensional basis (3.2.3), so that both formulations are equivalent in the 2D case.
An extension of Yahata’s approach to 3D formulation would require replacement of the stream
function by a vector potential, which would complicate the formulation.
Suslov & Paolucci (2002) also solved similar convection problem in a cavity with
coordinate functions (3.2.3). For the projection system they used the same functions multiplied
by the Chebyshev weight, which made the Gram matrix sparser and allowed to use the fast
Fourier transform (FFT) to evaluate non-linear terms of the dynamical system. The whole
approach was used for straightforward integration in time, but did not exhibit much advantage
compared to other methods and, to the best of the author’s knowledge, had no further
continuation.
10. To conclude: what else can be done?
To discuss further possible implementations of this Galerkin approach we mention that
with the present growth of computational power and state-of-the-art methods of numerical linear
43
algebra, the solution of two-dimensional and quasi-two-dimensional problems became feasible,
and sometimes more efficient, with lower order methods. A very popular methodology of turning
a time-stepping code into a steady state / stability solver can be found in Boronska & Tuckerman
(2010a,b) and Tuckerman et al (2018). Another possible methodology together with several
examples are given in Gelfgat (2007a,b). For these problems the Galerkin approach can be more
suitable for weakly non-linear bifurcations analysis. It is not clear, however, whether results
applicable only for small supercriticality will justify the effort.
One of possible applications of the method is consideration of fully three-dimensional
flows, steady and unsteady, in axisymmetric domains. In these problems the bases (3.4.5) and
(3.4.6) can be combined with the Fourier decomposition in the circumferential direction, so that
the Gram matrices will separate for each Fourier mode and thus not be too large, so that they will
be easily inverted. Finally, one obtains an ODE system, where equations corresponding to
different Fourier modes will be coupled via the non-linear terms. This system allows for
computation of steady states, path-following, stability analysis and time-dependent calculations
(see, e.g., Boronska & Tuckerman, 2010a,b).
Possibly, the most challenging task would be to develop a fully three-dimensional solver
for flow in a three-dimensional rectangular box. This would require orthogonalization of the
whole set of bases (3.3.2), (3.3.3), and (3.3.6) with subsequent effective treatment of the non-
linear terms. If successful, this approach would allow one to have steady state, stability, and
time-dependent solvers within a single computational model, with which very complicated flows
can be studied. It should be mentioned here that Krylov subspace based solvers are sometimes
thought to be applicable only to sparse matrices, for which matrix-vector products can be quickly
evaluated. However, matrix-vector products are also fast for the method that we have described,
even though all the related matrices are densely filled.
As mentioned in the very beginning of this paper, the Galerkin approach that we have
described is limited to simple domains, which must be curvilinear rectangles, in other words,
regions bounded by coordinate surfaces. This is a very stringent restriction since it excludes a
very big set of important problems, in which the boundaries have a more complicated shape.
Another restriction for implementation of this method is flows with deformable interfaces. One
of the ways to solve such problems on fixed grids is the immersed boundary method and/or the
diffuse interface approach (not described here). Implementation of these approaches for the
44
spectral method would require a good approximation of the delta function, which can be difficult
to do using smooth polynomials.
Finally, we emphasize a technique for flow visualization made in Gelfgat (2014). This is
applicable to flows calculated by any of numerical method, and can be very helpful for
understanding the topology of complicated three-dimensional flows.
References:
Batina J, Blancher S, Amrouche C, Batchi M, Creff R (2009) Convective heat transfer
augmentation through vortex shedding in sinusoidal constricted tube, Int. J. Numer. Meths Heat
Fluid Flow, 19: 374-395.
Blackburn HM, Lopez JM (2000) Symmetry breaking of the flow in a cylinder driven by a
rotating endwall Phys. Fluids 12: 2698–2701.
Bistrian DA, Dragomirescu IA, Muntean S, Topor M (2009) Numerical methods for convective
hydrodynamic stability of swirling flows, Proc. 13th WSEAS Int. Conf. on Systems, 283-288.
Borget V, Bdéoui F, Soufiani A, Le Quéré P (2001) The transverse instability in a differentially
heated vertical cavity filled with molecular radiating gases. I. Linear stability analysis, Phys.
Fluids, 13:1492-1507.
Borońska K Tuckerman LS (2010a) Extreme multiplicity in cylindrical Rayleigh-Bénard
convection. I. Time dependence and oscillations. Phys. Rev. E, 81: 036320.
Borońska K, Tuckerman LS (2010b) Extreme multiplicity in cylindrical Rayleigh-Bénard
convection. II. Bifurcation diagram and symmetry classification. Phys. Rev. E, 81: 036321.
Boyd JP (2000) Chebyshev and Fourier Spectral Methods. Dover Publications, New York.
Buffat M, Le Penven L (2011) A spectral fictitious domain method with internal forcing for
solving elliptic PDEs, J. Comput. Phys., 230: 2433-2450.
Canuto C, Quarteroni A, Hussaini MY, Zhang TA (2006) Spectral Methods. Fundamentals in
Single Domains. Springer, Berlin Heidelberg New York.
Dean WR (1928) ‘Fluid flow in a curved channel, Proc. R. Soc. London, Ser.A 121: 402-420.
Dumas G, Leonard A (1994) A divergence free spectral expansion method for three-dimensional
flows in spherical gap geometries, J. Comput. Phys., 11: 205-219.
Erenburg V, Gelfgat AY, Kit E, Bar-Yoseph PZ, Solan A (2003) Multiple states, stability,
bifurcations of natural convection in rectangular cavity with partially heated vertical walls, J.
Fluid Mech., 492: 63-89.
45
Escudier MP (1984) Observation of the flow produced in a cylindrical container by a rotating
endwall, Exp. Fluids, 2:189-196.
Fletcher CAJ (1984) Computational Galerkin Methods. Springer, New York.
Ganske A, Gebhardt T, Grossmann S (1994) Modulation effects along stability border in Taylor–
Couette flow, Phys. Fluids, 6: 3823-3832.
Gelfgat AY (1988) Instability and oscillatory supercritical regimes of free convection in a
laterally heated square cavity, PhD (Cand. Sci.) thesis. Latvian State University, Riga, Latvia.
Gelfgat AY, Tanasawa I (1994) Numerical analysis of oscillatory instability of buoyancy
convection with the Galerkin spectral method, Numer. Heat Transfer. Part A: Applications, 25:
627-648.
Gelfgat AY, Bar-Yoseph PZ, Solan A (1996a) Stability of confined swirling flow with, without
vortex breakdown., J. Fluid Mech., 311: 1-36.
Gelfgat AY, Bar-Yoseph PZ, Solan A (1996b) Steady states, oscillatory instability of swirling
flow in a cylinder with rotating top, bottom, Phys. Fluids, 8: 2614-2625.
Gelfgat AY, Bar-Yoseph PZ, Yarin AL (1997) On oscillatory instability of convective flows at
low Prandtl number, J. Fluids Eng., 119: 823-830.
Gelfgat AY, Bar-Yoseph PZ, Yarin AL (1999a) Stability of multiple steady states of convection
in laterally heated cavities, J. Fluid Mech., 388: 315-334.
Gelfgat AY, Bar-Yoseph PZ, Yarin AL (1999b) Non-Symmetric convective flows in laterally
heated rectangular cavities, Int. J. Computational Fluid Dynamics, 11: 261-273.
Gelfgat AY, Bar-Yoseph PZ, Solan A, Kowalewski T (1999c) An axisymmetry- breaking
instability in axially symmetric natural convection, Int. J. Transport Phenomena, 1: 173-190.
Gelfgat AY (1999) Different modes of Rayleigh-Bénard instability in two-, three-dimensional
rectangular enclosures, J. Comput. Phys. 156: 300-324.
Gelfgat AY, Bar-Yoseph PZ, Solan A (2000) Axisymmetry breaking instabilities of natural
convection in a vertical Bridgman growth configurations, J. Cryst. Growth, 220: 316-325.
Gelfgat AY (2001) Two-, three-dimensional instabilities of confined flows: numerical study by a
global Galerkin method, Computational Fluid Dynamics Journal, 9: 437-448.
Gelfgat AY, Bar-Yoseph PZ, Solan A. (2001a) Three-dimensional instability of axisymmetric
flow in a rotating lid - cylinder enclosure, J. Fluid Mech., 438: 363-377.
Gelfgat AY, Bar-Yoseph PZ, Solan A (2001b) Effect of axial magnetic field on three-
dimensional instability of natural convection in a vertical Bridgman growth configuration, J
Cryst.Growth, 230: 63-72.
46
Gelfgat AY, Bar-Yoseph PZ (2001c) The effect of an external magnetic field on oscillatory
instability of convective flows in a rectangular cavity, Phys. Fluids, 13: 2269-2278.
Gelfgat AY, Yarin AL, Bar-Yoseph PZ (2001d) Three-dimensional instability of a two-layer
Dean flow, Phys. Fluids, 13: 3185-3195.
Gelfgat AY (2002) Three-dimensionality of trajectories of experimental tracers in a steady
axisymmetric swirling flow: effect of density mismatch, Theor. Comput. Fluid Dyn., 16: 29-41.
Gelfgat AY (2004) Stability, slightly supercritical oscillatory regimes of natural convection in a
8:1 cavity: solution of benchmark problem by a global Galerkin method, Int. J. Numer. Meths.
Fluids, 44: 135-146.
Gelfgat AY, Bar-Yoseph PZ (2004) Multiple solutions, stability of confined convective, swirling
flows – a continuing challenge, Int. J. Numer. Meth. Heat, Fluid Flow, 14: 213-241.
Gelfgat YM, Gelfgat AY (2004) Experimental, numerical study of rotating magnetic field driven
flow in cylindrical enclosures with different aspect ratios, Magnetohydrodynamics, 40: 147-160.
Gelfgat AY (2005) On three-dimensional instability of a traveling magnetic field driven flow in a
cylindrical container, J. Crystal Growth, 279: 276-288.
Gelfgat AY (2006) Implementation of arbitrary inner product in global Galerkin method for
incompressible Navier-Stokes equation, J. Comput. Phys., 211: 513-530.
Gelfgat AY (2007a) Stability of convective flows in cavities: solution of benchmark
problems by a low-order finite volume method, Int. J. Numer. Meths. Fluids, 53: 485-506.
Gelfgat AY (2007b) Three-dimensional instability of axisymmetric flows: solution of
benchmark problems by a low-order finite volume method, Int. J. Numer. Meths. Fluids, 54:
269-294.
Gelfgat AY (2014) Visualization of three-dimensional incompressible flows by quasi-two-
dimensional divergence-free projections, Computers & Fluids, 97: 143-155.
Godeferd FS, Lollini L (1999) Direct numerical simulations of turbulence with confinement
and rotation, J. Fluid Mech., 393: 257-308.
Grants I, Gerbeth G (2001) Stability of axially symmetric flow driven by a rotating magnetic
field in a cylindrical cavity, J. Fluid Mech., 431: 407-426.
Gresho PM, Sani RL (1987) On pressure boundary conditions for the incompressible Navier–
Stokes equations, Int. J. Numer. Meth. Fluids, 7: 1111-1145.
Hassard BD, Kazarinoff ND, Wan YH (1981) Theory and Applications of Hopf Bifurcation.
London, Math. Soc. Lecture Note Series, vol.41.
Holte S (1983) Numerical experiments with a three-dimensional model of an enclosed basin,
Continental Shelf Research, 2: 301-315.
47
Iwatsu R (2005) Numerical study of flows in a cylindrical container with rotating bottom and top
free surface, J. Phys. Soc. Japan, 74: 333:344.
Kerr R (1996) Rayleigh number scaling in numerical convection, J. Fluid Mech., 310: 139-179.
Marques F, Lopez JM (2001). Precessing vortex breakdown mode in an enclosed cylinder flow.
Phys. Fluids, 13: 1679–1682.
Marques F, Lopez JM, Shen J (2002). Mode interactions in an enclosed swirling flow: A double
Hopf bifurcation between azimuthal wavenumbers 0 and 2. J. Fluid Mech., 455: 263–281.
Marques F., Gelfgat AY, Lopez JM (2003) Tangent double Hopf bifurcation in a differentially
rotating cylinder flow, Phys. Rev. E, 68: 06310-1 – 06310-13 .
Mason JC, Handscomb DC (2003) Chebyshev Polynomials, CRC Press, London.
Meseguer A, Mellibovsky F (2007) On a solenoidal Fourier–Chebyshev spectral method for
stability analysis of the Hagen–Poiseuille flow, Appl. Numer. Math., 920-938.
Moser RD, Moin P, Leonard A (1983) A spectral numerical method for the Navier-Stokes
equations with applications to Taylor-Couette flow, J. Comput. Phys., 52: 524-544.
Nore C, Tuckerman LS, Daube O, Xin S (2003) The 1:2 mode interaction in exactly counter-
rotating von Karman swirling flow, J. Fluid Mech., 477: 51–88.
Orszag SA (1971a) Galerkin approximations to flows within slabs, spheres, and cylinders, Phys.
Rev. Lett., 26: 1100-1103.
Orszag SA (1971b) Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech,
50: 689-703.
Pasquarelli F (1991) Domain decomposition for spectral approximation to Stokes equations via
divergence-free functions, Appl. Umr. Math., 8: 493-51.
Paszkowski S (1975) Numerical Applications of Chebyshev Polynomials and Series, PWN,
Warsaw (in Polish).
Picardo JR, Garg P, Pushpavanam S (2015) Centrifugal instability of stratified two-phase flow in
a curved channel, Phys. Fluids, 27: 054106.
Rempfer D (2006) On boundary conditions for incompressible Navier-Stokes problems, Appl.
Mech. Rev., 59: 107-125.
Rubinov A, Erenburg V, Gelfgat AY, Kit E, Bar-Yoseph PZ, Solan A (2004) Three-dimensional
instabilities of natural convection in a vertical cylinder with partially heated sidewalls, J. Heat
Transfer, 126: 586-599.
Sørensen JN, Naumov I, Mikkelsen R (2006) Experimental investigation of three-dimensional
flow instabilities in a rotating lid-driven cavity, Exp. Fluids, 41: 425-440.
48
Sørensen JN, Gelfgat AY, Naumov I, Mikkelsen R (2009) Experimental and numerical results on
three-dimensional instabilities in a rotating disk–tall cylinder flow, Phys. Fluids, 21: 054102.
Suslov SA, Paolucci S (2002) A Petrov-Galerkin method for flows in cavities: enclosure of
aspect ratio 8, Int. J. Numer. Meths. Fluids, 40: 999-1007.
Tuckerman LS, Langham J, Willis A., (2018) Stokes preconditioning in Channelflow and
Openpipeflow for steady states and traveling waves. In: <this book>.
Uhlmann M, Nagata M (2006) Linear stability of flow in an internally heated rectangular duct, J.
Fluid Mech., 551: 387-404.
Wan X,Yu H (2017) dynamic-solver–consistent minimum action method: With an application to
2D Navier–Stokes equations, J. Comput. Phys., 331: 209-226.
Yahata H (1999) Stability analysis of natural convection in vertical cavities with lateral heating,
J. Phys. Soc. Japan, 68: 446-460.
Yahata H (2001) Stability analysis of natural convection evolving along a vertical heated plate, J.
Phys. Soc. Japan, 70: 11-130.
Yang WM (1997) Stability of viscoelastic fluids in a modulated gravitational field, Int. J. Heat
Mass Transfer, 40: 1401-1410.
Yueh CS, Weng CI (1996) Linear stability analysis of plane Couette flow with viscous heating,
Phys. Fluids, 8:1802-1813.
Zebib A (1984) A Chebyshev method for the solution of boundary value problems, J.
Comput. Phys., 53: 443-455.
49
Appendix A. Shifted Chebyshev polynomials and some of their useful properties
Shifted Chebyshev polynomials of the first and the second type shifted onto the interval
01 are defined as
()=[ (21)], ()=[() ()]
[ ()] , 0 1 (A1)
The two systems {()}
and {()}
form bases in [0,1] and are connected via the
relation ()= 2(+ 1)() , (A2)
which resembles connection between sine and cosine. Values of the polynomials and their
derivatives in the points = 0 and = 1 that are needed to define basis functions for different
boundary conditions are
(0)=(1), (1)= 1 (A3)
(0)=(1)(+ 1), (1)=+ 1 (A4)
(0)=(1)2, (1)= 2 (A5)
(0)=(1)(+ 1)(+ 2)3
, (1)=(+ 1)(+ 2)3
(A6)
For the following we assume that for > 0, ()=() and ()=()=
(). To evaluate inner products we need to decompose the polynomial derivatives and the
polynomials products into polynomial sums. For the multiplication of a polynomial by a
polynomial we have
()()=
()+() (A7)
()()=
()+() (A8)
()()= ()
(A9)
The derivatives can be represented as Chebyshev series as
()= 2!(+)+
+1
()
[()
]
(A10)
()= 2(+ 1)!++ 1
+ 1 +
+ 1 ()
[()
]
(A11)
=1
2, = 1
50
For example, ()= 4()
[()
]
(A12)
Here
is the binomial coefficient. After the basis functions are built, the Galerkin projections
can be computed with the unity weight
,=()()
, (A13)
or with the Chebyshev weight
,=()
()()
, (A14)
or with an arbitrary weight. For example, in Gelfgat (2005) we used
,=()()()
, 0 < < 1 (A15)
All the inner products needed to complete the Galerkin procedure can be evaluated analytically if
either the unity or Chebyshev weight is implied. The following relations (base products) are
needed for that
(),()=
[1 + (1)]
+
(A16)
(),()= (A17)
And is the Kronecker symbol. The relation (A2), (A7)-(A16) allow one to reduce all the
inner products to the above ones. In the case of arbitrary inner product the base products
(),() must be evaluated numerically, which is usually done using the Gauss
quadrature. Then all the other products can be expressed as sums using relations (A2) and (A7)-
(A11). Additionally, for evaluation of inner products in orthogonal curvilinear coordinates, one
may need the following relation
= 22
()
(A18)
Finally, to calculate the shifted Chebyshev polynomials in a point, the following recurrent
formulae can be used ()=(42)()() (A19)
()=(42)()() (A20)
Further details can be found in the books of Paszkowski (1975) and Mason & Handscomb
(2003).
