Alternative approach to implement the effect of coalescence in the Stochastic Field Method assuming a nucleus number of 3. To be consistent in the volume fraction another field N + 1 needs to be introduced temporarily. 

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... α i is the volume fraction of field i. A task strongly related to two phase flow is how to treat the effect of coalescence. The number of bubbles including coalescence can be modelled by a correlation, see [11]. Alternatively it can be modelled solving an additional transport equation for the bubble number or for the bubbles' interfacial area respectively [12]. The Stochastic Field Method can adopt one of these methods and derive the bubble size from the volume fraction assuming spherical bubbles. N different volume fractions within the stochastic fields lead to N different bubble sizes. Another approach to treat coalescence within the Stochastic Field Method is to allow an interaction among the fields: At the first timestep of the simulation the nucleus number is estimated by the user. Again, assuming spherical bubbles and using the volume fraction within each stochastic field a bubble size for each field is derived. Exemplary bubble sizes for each field are given in row 1 of Figure 3 for a nucleus number of 3. If coalescence occures between field N − 1 and field N -this information could be taken from a PDF of the distance in between of two bubbles or a Lagrangian model -the resulting 'physical configuration post coalescence' is shown in the second row of Figure 3: Field N − 1 contains bigger bubbles and field N is empty. Although this implementation would be physical as it takes into concideration single bubbles it violates the principle of the Stochastic Field Method using N samples to approximate the PDF, cmp. Fig. 1. Thus, field N needs to contain a new sample (red in Fig. 3) previously not taken into consideration. This results in an increase of the average volume fraction because in comparancy to the situation prior to coalescence the sum of volume fractions within the fields increases whereas the number of fields remains constant, cp. Eq. 2. To avoid this, a field N + 1 with zero volume fraction can virtually be introduced, cp. Fig. 3 last row. Hereby the effect of coalescence is implemented considering an interaction in between of fields while not increasing the average volume ...

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... α i is the volume fraction of field i. A task strongly related to two phase flow is how to treat the effect of coalescence. The number of bubbles including coalescence can be modelled by a correlation, see [11]. Alternatively it can be modelled solving an additional transport equation for the bubble number or for the bubbles' interfacial area respectively [12]. The Stochastic Field Method can adopt one of these methods and derive the bubble size from the volume fraction assuming spherical bubbles. N different volume fractions within the stochastic fields lead to N different bubble sizes. Another approach to treat coalescence within the Stochastic Field Method is to allow an interaction among the fields: At the first timestep of the simulation the nucleus number is estimated by the user. Again, assuming spherical bubbles and using the volume fraction within each stochastic field a bubble size for each field is derived. Exemplary bubble sizes for each field are given in row 1 of Figure 3 for a nucleus number of 3. If coalescence occures between field N − 1 and field N -this information could be taken from a PDF of the distance in between of two bubbles or a Lagrangian model -the resulting 'physical configuration post coalescence' is shown in the second row of Figure 3: Field N − 1 contains bigger bubbles and field N is empty. Although this implementation would be physical as it takes into concideration single bubbles it violates the principle of the Stochastic Field Method using N samples to approximate the PDF, cmp. Fig. 1. Thus, field N needs to contain a new sample (red in Fig. 3) previously not taken into consideration. This results in an increase of the average volume fraction because in comparancy to the situation prior to coalescence the sum of volume fractions within the fields increases whereas the number of fields remains constant, cp. Eq. 2. To avoid this, a field N + 1 with zero volume fraction can virtually be introduced, cp. Fig. 3 last row. Hereby the effect of coalescence is implemented considering an interaction in between of fields while not increasing the average volume ...

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... α i is the volume fraction of field i. A task strongly related to two phase flow is how to treat the effect of coalescence. The number of bubbles including coalescence can be modelled by a correlation, see [11]. Alternatively it can be modelled solving an additional transport equation for the bubble number or for the bubbles' interfacial area respectively [12]. The Stochastic Field Method can adopt one of these methods and derive the bubble size from the volume fraction assuming spherical bubbles. N different volume fractions within the stochastic fields lead to N different bubble sizes. Another approach to treat coalescence within the Stochastic Field Method is to allow an interaction among the fields: At the first timestep of the simulation the nucleus number is estimated by the user. Again, assuming spherical bubbles and using the volume fraction within each stochastic field a bubble size for each field is derived. Exemplary bubble sizes for each field are given in row 1 of Figure 3 for a nucleus number of 3. If coalescence occures between field N − 1 and field N -this information could be taken from a PDF of the distance in between of two bubbles or a Lagrangian model -the resulting 'physical configuration post coalescence' is shown in the second row of Figure 3: Field N − 1 contains bigger bubbles and field N is empty. Although this implementation would be physical as it takes into concideration single bubbles it violates the principle of the Stochastic Field Method using N samples to approximate the PDF, cmp. Fig. 1. Thus, field N needs to contain a new sample (red in Fig. 3) previously not taken into consideration. This results in an increase of the average volume fraction because in comparancy to the situation prior to coalescence the sum of volume fractions within the fields increases whereas the number of fields remains constant, cp. Eq. 2. To avoid this, a field N + 1 with zero volume fraction can virtually be introduced, cp. Fig. 3 last row. Hereby the effect of coalescence is implemented considering an interaction in between of fields while not increasing the average volume ...

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... α i is the volume fraction of field i. A task strongly related to two phase flow is how to treat the effect of coalescence. The number of bubbles including coalescence can be modelled by a correlation, see [11]. Alternatively it can be modelled solving an additional transport equation for the bubble number or for the bubbles' interfacial area respectively [12]. The Stochastic Field Method can adopt one of these methods and derive the bubble size from the volume fraction assuming spherical bubbles. N different volume fractions within the stochastic fields lead to N different bubble sizes. Another approach to treat coalescence within the Stochastic Field Method is to allow an interaction among the fields: At the first timestep of the simulation the nucleus number is estimated by the user. Again, assuming spherical bubbles and using the volume fraction within each stochastic field a bubble size for each field is derived. Exemplary bubble sizes for each field are given in row 1 of Figure 3 for a nucleus number of 3. If coalescence occures between field N − 1 and field N -this information could be taken from a PDF of the distance in between of two bubbles or a Lagrangian model -the resulting 'physical configuration post coalescence' is shown in the second row of Figure 3: Field N − 1 contains bigger bubbles and field N is empty. Although this implementation would be physical as it takes into concideration single bubbles it violates the principle of the Stochastic Field Method using N samples to approximate the PDF, cmp. Fig. 1. Thus, field N needs to contain a new sample (red in Fig. 3) previously not taken into consideration. This results in an increase of the average volume fraction because in comparancy to the situation prior to coalescence the sum of volume fractions within the fields increases whereas the number of fields remains constant, cp. Eq. 2. To avoid this, a field N + 1 with zero volume fraction can virtually be introduced, cp. Fig. 3 last row. Hereby the effect of coalescence is implemented considering an interaction in between of fields while not increasing the average volume ...

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... assuming spherical bubbles and using the volume fraction within each stochastic field a bubble size for each field is derived. Exemplary bubble sizes for each field are given in row 1 of Figure 3 for a nucleus number of 3. If coalescence occures between field N − 1 and field N -this information could be taken from a PDF of the distance in between of two bubbles or a Lagrangian model -the resulting 'physical configuration post coalescence' is shown in the second row of Figure 3: Field N − 1 contains bigger bubbles and field N is empty. Although this implementation would be physical as it takes into concideration single bubbles it violates the principle of the Stochastic Field Method using N samples to approximate the PDF, cmp. ...

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... assuming spherical bubbles and using the volume fraction within each stochastic field a bubble size for each field is derived. Exemplary bubble sizes for each field are given in row 1 of Figure 3 for a nucleus number of 3. If coalescence occures between field N − 1 and field N -this information could be taken from a PDF of the distance in between of two bubbles or a Lagrangian model -the resulting 'physical configuration post coalescence' is shown in the second row of Figure 3: Field N − 1 contains bigger bubbles and field N is empty. Although this implementation would be physical as it takes into concideration single bubbles it violates the principle of the Stochastic Field Method using N samples to approximate the PDF, cmp. ...

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... in the second row of Figure 3: Field N − 1 contains bigger bubbles and field N is empty. Although this implementation would be physical as it takes into concideration single bubbles it violates the principle of the Stochastic Field Method using N samples to approximate the PDF, cmp. Fig. 1. Thus, field N needs to contain a new sample (red in Fig. 3) previously not taken into consideration. This results in an increase of the average volume fraction because in comparancy to the situation prior to coalescence the sum of volume fractions within the fields increases whereas the number of fields remains constant, cp. Eq. 2. To avoid this, a field N + 1 with zero volume fraction can ...

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... taken into consideration. This results in an increase of the average volume fraction because in comparancy to the situation prior to coalescence the sum of volume fractions within the fields increases whereas the number of fields remains constant, cp. Eq. 2. To avoid this, a field N + 1 with zero volume fraction can virtually be introduced, cp. Fig. 3 last row. Hereby the effect of coalescence is implemented considering an interaction in between of fields while not increasing the average volume ...