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... 4.2. Cylindrical Couette Flow The next test was conducted for cylindrical Couette flow, which is a flow between two coaxial rotating cylinders. There is a well-known stationary solution for this flow=-=[17]-=-: �=ar� b r (35) where a= � 2 R 2 2 R2−R1 2 , b=−� 2 R 2 1 R 2 2 −R1 R 2 2 (36) Here � 2 is the velocity of the outer cylinder (if the inner one is not rotating), R 1, R 2 are inner and outer cylinder...

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...s: �=∑ i f i , � �=∑ i f i c i (3) Note that lattice vectors and their weights are constructed to satisfy symmetry conditions [1] and to recover density and velocity for continuous Boltzmann dynamics =-=[12]-=-. Typical lattice stencil for a two-dimensional space is the D2Q9 one, which contains of 9 lattice vectors with the following coordinates [2] � 0 0 ∣�1∣ 0 0 �1 ∣−1∣ 0 0 −1 ∣�1 �1 ∣−1 �1 ∣−1 −1 ∣�1 −1�...

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...s numerically stable, accurate and local; it has a good physical background. Also, we provide theoretical justifications of a popular Zou-He boundaries technique and an isothermal boundaries algorithm=-=[10]-=-. These 4justifications are based on the maximum entropy method derived. Finally, while conducting numerical benchmarks, typical straight boundaries algorithm (Zou-He) was compared to a typical curve...

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...just plane boundary conditions), Chen-Martinez boundary nodes (which support curved boundary conditions, but need specific treatment for curved boundaries, so they were used just for plane walls) [5],=-=[8]-=-,[10]. 4.1. Poiseuille Flow We consider an academic two-dimensional Poiseuille flow within the following setting. Flow is force-driven to prevent pressure and density changes and to make the simulatio...

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...nt to specify either velocity or density (not both of them) due to the Navier-Stokes equation properties. A ubiquitous approach for no-slip boundaries implementation is the usage of bounce-back nodes =-=[1]-=-. Also, several specific methods exist for constant density and constant velocity boundaries modeling on straight walls [4]. An accurate method for constant velocity boundaries simulation on curved wa...

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...entioned above, are measured in lattice units [13]. LBM method analysis and linking it to the Navier-Stokes equation are usually performed through one of the following techniques: 1. Taylor expansion =-=[14]-=- 2. asymptotic analysis [15] 3. Chapman-Enskog expansion [1]. 83. Maximum Entropy Boundaries In this section maximum entropy boundaries method will be explained. 3.1. Method Idea F B E C 6 C 2 C 5 D ...

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...sk of implementing boundary conditions to make them correspond to the macroscopic constraints. The most wide-spread boundaries are solid no-slip ones, constant velocity or constant density boundaries =-=[4]-=-. Note that it is always sufficient to specify either velocity or density (not both of them) due to the Navier-Stokes equation properties. A ubiquitous approach for no-slip boundaries implementation i...

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...ial velocity is u , pressure is p , membrane velocity, directed outside, is V , cylinder radius is a , non-dimensional radial coordinate �=r 2 /a 2 . A general solution for such a problem has the form=-=[20]-=- �=V f �� �/���� u=u 0 f ' ����1−2V z/a� p= p 0 ��� A z 2 �Bz���V 2 � 2 f ' / R− f 2 /� � (37) where u 0 is a characteristic velocity, R=V a /� is suction Reynolds number (don't mix it with the main R...

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...rmined during the solution of the boundary-value problem (38) due to sufficient boundary conditions. 23The approximate solution to this problem, obtained with the perturbation theory, can be found in=-=[22]-=-,[23] f ���=2�−� 2 � R 36 � 4� −9�2 �6� 3 −� 4 �, K =−2� 3 2 R This solution is applicable, when wall Reynolds number R is in the range [−2 ;2] . A D3Q19 scheme was utilized in the simulation. Cylinde...

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... time t . These probabilities are normalized by mass of the fluid. � is the mass relaxation time, 6which is related to the kinematic viscosity as 2 �=c s ��−1/2� , where cs=1/��3� is the sound speed =-=[11]-=-. Equilibrium particle population in direction i is eq f i =wi �[1�3�⋅c i� 9 2 ��⋅c i� 2 − 3 2 � 2 ] , (2) where macroscopic density in the given node is � , macroscopic velocity is � . Lattice vector...

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... in lattice units [13]. LBM method analysis and linking it to the Navier-Stokes equation are usually performed through one of the following techniques: 1. Taylor expansion [14] 2. asymptotic analysis =-=[15]-=- 3. Chapman-Enskog expansion [1]. 83. Maximum Entropy Boundaries In this section maximum entropy boundaries method will be explained. 3.1. Method Idea F B E C 6 C 2 C 5 D C 3 O C 0 C 1 A C 7 C 4 C 8 ...

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...d during the solution of the boundary-value problem (38) due to sufficient boundary conditions. 23The approximate solution to this problem, obtained with the perturbation theory, can be found in[22],=-=[23]-=- f ���=2�−� 2 � R 36 � 4� −9�2 �6� 3 −� 4 �, K =−2� 3 2 R This solution is applicable, when wall Reynolds number R is in the range [−2 ;2] . A D3Q19 scheme was utilized in the simulation. Cylinder len...

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...vectors ∣c i∣ 2 =1 , w i =1/36 for vectors ∣c i∣ 2 =2 . 7An important point in the LBM framework is specific units conversion: all the physical values, mentioned above, are measured in lattice units =-=[13]-=-. LBM method analysis and linking it to the Navier-Stokes equation are usually performed through one of the following techniques: 1. Taylor expansion [14] 2. asymptotic analysis [15] 3. Chapman-Enskog...

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...stem widths (Y axis) were obtained in the first set of experiments and errors between these solutions and expected solutions were measured. Solutions error was computed according to the Maier L 2 norm=-=[16]-=- in the space of possible system states by the following formula: 17�= � ∑ ∣u nodes theor −u exp ∣ 2 � ∑ ∣u nodes theor ∣ 2 (34) Numerical simulations were stopped when the difference by the given no...

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...e operating according to the Darcy's law[18-22]. Fluid velocity through membranes is assumed to be constant and predefined for simplicity. Despite there is an exact analytical solution for such a case=-=[21]-=-, it turns out that an approximate solution is simpler and easier for analysis and implementation. The following variables are introduced: axial velocity is � , radial velocity is u , pressure is p , ...

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...er density and velocity for continuous Boltzmann dynamics [12]. Typical lattice stencil for a two-dimensional space is the D2Q9 one, which contains of 9 lattice vectors with the following coordinates =-=[2]-=- � 0 0 ∣�1∣ 0 0 �1 ∣−1∣ 0 0 −1 ∣�1 �1 ∣−1 �1 ∣−1 −1 ∣�1 −1� (4) Lattice weights for this scheme are the following: w i =4/9 for a vector with ∣c i∣ 2 =0 , w i =1/9 for vectors ∣c i∣ 2 =1 , w i =1/36 f...