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Chapter 5 presents the final universal form of the discretized governing equations for all flow conservations (e.g., mass, momentum, energy).
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Example Project: SemiLagrangianbased PISO method for fast and accurate indoor modelling.
The demand for fast engineering modeling has led to various means and efforts to reduce the cost of CFD techniques. Some of these efforts include: developing simplified turbulence models such as zeroequation models (Chen and Xu
1998); reforming solution algorithms for pressurevelocity decoupling such as Pressure Implicit with Splitting of Operator (PISO) (Issa
1986) and projection methods (Chorin
1967); utilizing coarse grids (Mora et al.
2003; Wang and Zhai
2012); and employing computer hardware technology such as Graphics Processing Unit (GPU) (Cohen and Molemake
2009) and parallel/multiprocessor supercomputers. Although the rapid development of computer hardware provides more powerful computing capacity, it does not address the challenge fundamentally.
Background:
Fast fluid dynamics (FFD) is a method widely used in weather prediction and atmospheric flow study (Robert
1981; Staniforth and Côté
1991). It solves the NavierStokes (NS) equations with a timeadvancement scheme and a semiLagrangian (SL) scheme. For instance, Foster and Metaxas (
1996,
1997) implemented the projection method (Chorin
1967) to simulate the 3D motion of hot, turbulent gas using a relatively coarse grid. Stam (
1999) proposed using semiLagrangian advection and fast Fourier transformation to speed up the computation to a realtime or fasterthanrealtime level. Zuo and Chen (
2009) first applied this operator splitting algorithm to 2D indoor environment modeling, improved the sequence of operators, tested higher orders of differencing schemes, and evaluated the accuracy levels. Zuo et al. (
2010,
2012) further improved the accuracy of FFD by using the finite volume method, mass conservation correction, and a hybrid interpolation scheme. Jin et al. (
2012,
2013,
2015) extended FFD to the solution of threedimensional airflow. Liu et al. (
2016) implemented FFD in OpenFOAM (
2007) with unstructured mesh, enabling the practical application of the algorithm. Even though FFD significantly accelerates the computation, its accuracy is still far from satisfaction. This study attempts to combine the semiLagrangian scheme with a PISO solver with the goal to increase the computation speed of PISO but without losing the accuracy (Xue et al.
2016).
Simulation Details:
A fully implicit algorithm is unconditionally stable and has no Courant–Friedrichs–Lewy (CFL) restriction (Issa
1986), thus it is commonly used in CFD. However, in solving the momentum equations numerically, the advection term is fundamentally different from others because it brings significant nonlinearity. SemiLagrangian scheme (Courant et al.
1952) shows potential for resolving the dilemma. The idea of semiLagrangian scheme was originated from the advection of scalar, but it can be directly applied to vector as well.
SemiLagrangian Advection
The Lagrangian method treats the continuum as a particle system. Each point in the fluid is labeled as a separate particle. From the perspective of such particles, the observed value of (i.g., density, temperature, etc.) will remain the same within the lapse of time. The semiLagrangian scheme follows the procedure as described in Fig.
8.5 to obtain the observed value of next time step. An existing velocity field of current time step
t provides the velocity at (any) point A of the grid. To predict point A’s value of next time step
\(t + \Delta t\), the semiLagrangian method traces back to point A’s upstream location B
\((\overrightarrow {AB} =  \vec{v} \cdot \Delta t)\) using the current velocity. This location B may not necessarily match an exact grid node. The surrounding values of current time step will be used to interpolate the value at this specific location. This value will then be kept and assigned to point A as its observed value of next time step. Since there is no CFL condition restriction, the time step and grid size used in a semiLagrangian scheme are usually large, which introduces large truncation error. Higher order numerical schemes can be used to improve the accuracy in the interpolation of the method.
An algorithm integrating semiLagrangian advection with the PISO algorithm is proposed as follows.
SemiLagrangian PISO Algorithm
Use the semiLagrangian advection ((
x,
−
t)) to obtain a first intermediate velocity field.
Step 1: SemiLagrangian Advection: Velocity
Intermediate velocity field
u
^{*} and initial pressure field
p
^{n} are used in the solution of the implicit momentum Eq. (
8.58) to yield a second intermediate velocity field
u
^{**}
Since this is using
p
^{n} instead of
p
^{**},
^{**} will not satisfy the continuity equation.
Step 2: Predictor Step: Velocity
An approximation of the velocity field
u
^{***} together with the corresponding new pressure field
p
^{***} are sought that satisfy the continuity equation
Step 3: First Corrector Step: Pressure
The momentum equation is then taken as
Equation (
8.60) subtracting Eq. (
8.58) yields
Take divergence for both sides of Eq. (
8.61), the velocity increment Eq. (
8.61) becomes the pressure increment equation to solve
p*** −
p
^{n} field
The updated pressure field (or pressure increment field) can be substituted into Eq. (
8.60) or Eq. (
8.61) to update the velocity field and produce the velocity field
u***.
Step 4: First Corrector Step: Velocity
A replication of Step 2 is conducted using the updated result from Step 3
u*** and after advection of the initial value
u*, with the newest pressure field
p****, yields an updated velocity field
where the explicit scheme in
\(v\nabla^{2} u^{***}\) is taken to operate on the
\(u^{***}\) field. Then
\(u^{****}\) corresponding with
\(p^{****}\) satisfies the continuity equation
Step 5:
Second Corrector Step: Pressure
Equation (
8.63) subtracting Eq. (
8.60) produces
After taking the divergence of both sides of Eq. (
8.65), together with the continuity equation
\(\nabla u^{****}\) and
\(\nabla u^{***}\), the velocity increment, Eq. (
8.65), yields the pressure increment equation to solve the
p
^{****} −
p
^{***} field:
The updated pressure field (or pressure increment field) can be plugged into Eq. (
8.63) or Eq. (
8.65) to update the velocity field and produce the velocity field
u
^{****}. More corrector steps can be used. However, the accuracy of two corrector steps is often adequate to approximate the exact solutions
u
^{n+1} and
p
^{n+1}.
Step 6: Second Corrector Step: Velocity
The temperature field is solved separately from the velocity field, in the PISO algorithm, although the procedure is similar. Considering the coupling between the temperature and velocity, the current study used the state equation of ideal gases to update the density of air, as shown in the steps below.
Use the semiLagrangian advection ((
x, −
t)) to obtain a first intermediate temperature field.
Step 7: SemiLagrangian Advection: Temperature
Intermediate temperature field
T
^{*} is used in the solution of the implicit energy Eq. (
8.68) to yield the temperature field
T
^{n+1}
Step 8: Corrector Step: Temperature
Update the density of air with the state equation of ideal gases.
Step 9: Update of Density
The proposed semiLagrangian PISO algorithm, without the corrector steps (Step 5 and Step 6), is similar to FFD except that it takes into consideration the pressure field from the previous time step. The FFD algorithm neglects the influence of pressure from the previous time step and assumes pressure is solely determined by the velocity field under the continuity restriction. In FFD, the advection term is completely separated from the rest of the momentum equation and is solved by using the semiLagrangian algorithm, which is faster and more stable compared to the conventional method of directly solving the advection equation. But the accuracy of PISO, theoretically and practically, has more advantages over FFD. The integrated algorithm (SLPISO) is expected to improve the accuracy of FFD without sacrificing much computing speed. The semiLagrangian advection algorithm is anticipated to largely reduce the computing cost of the direct solving of the advection term in the original PISO algorithm.
It is critical to evaluate the performance of the developed algorithm for both steady and unsteady problems. A liddriven cavity flow case and a mixing convection case in a confined space are used to evaluate and illustrate method performance. Figure
8.6 shows the liddriven cavity laminar flow under isothermal condition (Ghia et al.
1982). Figure
8.7 shows a 2D mixing convection case (Blay et al.
1992) with temperature impacts.
Results and Analysis:
The study compares the performance of four algorithms: SIMPLE, PISO, FFD and SLPISO. The mesh size is 50 × 50 and the time step size is 0.005 s. Figure
8.8a shows the predicted V
_{y} at line Y = 0.5 m. The results of PISO and SLPISO are almost identical. SIMPLE provides similar velocity magnitude while FFD obtains considerably different results. The results in this case reveal that SLPISO shares the same accuracy as PISO, with a slight deviation from experimental data. All of them provide better results than FFD. Figure
8.8b compares the computing costs. SIMPLE requires much more time. SLPISO has a similar speed as FFD. However, both of them are slower than PISO in this case, which will be explained later.
The study evaluates the algorithm with a 2D mixing convection case (Blay et al.
1992) that includes the temperature field. Experimental results were obtained from the literature, which were measured in a laboratory chamber of 1.04 m × 1.04 m × 0.7 m (x × y × z) equipped with a 18 mm wide inlet slot and a 24 mm wide outlet slot. The experiment produced a fairly good 2D flow at the central plate. The experiment measured wall temperatures and supply air conditions, respectively, as T
_{roof} = T
_{walls} = 15 °C, T
_{floor} = 35.5 °C, T
_{inlet} = 15 °C, V
_{inlet} = 0.57 m/s (normal to the inlet slot), as well as temperature, V
_{y} at the ten points along the middle line on the central plate (as shown in Fig.
8.7). This study uses the constant effective kinematic viscosity and heat transfer coefficient, namely one hundred times of the physical values, to consider the turbulence impact. The mesh size is 80 × 80 and the time step size is 0.005 s. Results in Fig.
8.9 demonstrate that the SLPISO algorithm provides similar results as the PISO method. FFD has a large disparity in temperature prediction. SLPISO has similar computational speed as FFD, while they are still slower than PISO.
To evaluate the transient simulation accuracy of SLPISO and FFD, a transient flow in the 2D mixing convection case is simulated. Since no transient experiment results exist for this case, the SIMPLE algorithm results are used as the reference for comparison. The “experimental” data is taken every five seconds from the SIMPLE prediction at the middle point of the test chamber. The study uses the mesh size of 80 × 80. The time step is varied from 0.005 to 0.08 s. As the time step increases, the transient results and the steady state results of SLPISO and FFD deviate, where SLPISO outperforms FFD in general (Fig.
8.12). FFD must use smaller time steps to obtain similar results as SLPISO.
Because of the inherent characteristics of the semiLagrangian scheme, SLPISO and FFD may not provide significant computing saving than the conventional CFD algorithms when the number of grid is relatively small. They gain their advantages when the number of grid is increased. Most engineering problems require more than onemillion grids to reach solutions of gridindependence, and thus FFD and SLPISO show great potential of fast simulation for these applications. This potential is further enhanced with the advantage of being able to use larger time steps for both SLPISO and FFD. SLPISO is slightly slower than FFD but with a higher accuracy especially for transient cases. SLPISO can adopt larger time steps than PISO and FFD to obtain accurate steady state results.
Discussions
This assignment will use a computational fluid dynamics (CFD) program to simulate the benchmark case of a computersimulated person (CSP) under a mixing ventilation condition (Fig.
8.13).
Indoor airflow and heat transfer simulation
Simplification of indoor object (person)
Comparison of simulation with experimental data.
Case Descriptions:
3D computational domain with dimensions of X × Y × Z = 2.44 × 1.2 × 2.46 m.
Air is supplied through the full crosssectional area at one end of the channel and leaves through two circular openings at the opposite end.
The circular exhaust openings have a diameter of 0.25 m and are located 0.6 m from the floor and the ceiling, respectively.
The CSPs are located 0.7 m from the inlet, centered on the xaxis.
The geometry of the CSP is based on an averagesized woman with a standing height of 1.7 m. When seated, the CSP has a height of 1.38 m. The surface area of the CSP is 1.52 m
^{2}.
Pick up a reasonable body size.
A uniform velocity profile of U = 0.2 m/s and T = 22 °C is applied to the opening. Inlet turbulence intensity k and dissipation rate ε values can be calculated based on the literature (ISSN 13957953 R0307).
A convective heat flow rate of 38.0 W is prescribed for the CSP corresponding to an activity level of approximately 1 Met (sedentary work).
Steady state w/o contaminant.
Other surfaces are adiabatic.
More case details and
experimental data can be found at:
http://homes.civil.aau.dk/pvn/cfdbenchmarks/csp_benchmark_test/.
Simulation Details:
Turbulence model: ReNormalization Group (RNG) k − ε model (Yakhot and Orszag
1986).
Convergence criterion: 0.1%.
Iteration: at least 1000 steps.
Grid: local refined grid with different total grid numbers.
Cases to Be Simulated:
KERNG model with
at least three different orders of grid numbers (e.g., 30 × 15 × 30, 45 × 23 × 45, 70 × 35 × 70).
Report:
Case descriptions: descriptions of the cases.
Simulation details: computational domain, grid cells, convergence status.
Figure of the best grid used (on X–Z and X–Y planes);
Figure of a typical convergence process recorded.
Result and analysis (only present the best results except for the 1st item).
Gridindependent solution: use one vertical pole at X = 1.69 m to compare and show the predicted velocity differences with different grids;
Figure of velocity contours at the middle height of the CSP;
Figure of airflow vectors at the middle height of the CSP;
Figure of temperature contours at the middle height of the CSP;
Figure of velocity contours at the central plane cross the CSP;
Figure of airflow vectors at the central plane cross the CSP;
Figure of temperature contours at the central plane cross the CSP;
Comparison of velocities along the three tested vertical poles at the central plane cross the CSP (experimentdot; simulationsolid line).
Conclusions (findings, CFD experience and lessons, etc.)

Background:

Simulation Details:
(1)
SemiLagrangian Advection
×
(2)
SemiLagrangian PISO Algorithm

Step 1: SemiLagrangian Advection: Velocity
$$\frac{{u^{*}  u^{n} }}{\Delta t} =  \left( {u^{*} \cdot \nabla } \right)u^{*} \Rightarrow u^{*} = u^{n} \left[ {P\left( {x,  \Delta t} \right)} \right]$$
(8.57)

Step 2: Predictor Step: Velocity
$$\frac{{u^{**}  u^{*} }}{\Delta t} = v\nabla^{2} u^{**}  \frac{1}{{\rho^{n} }}\nabla p^{n} + S^{u}$$
(8.58)

Step 3: First Corrector Step: Pressure
$$\nabla u^{***} = 0$$
(8.59)
$$\frac{{u^{***}  u^{*} }}{\Delta t} = v\nabla^{2} u^{**}  \frac{1}{{\rho^{n} }}\nabla p^{***} + S^{u}$$
(8.60)
$$\frac{{u^{***}  u^{**} }}{\Delta t} =  \frac{1}{{\rho^{n} }}\left( {\nabla p^{***}  \nabla p^{n} } \right)$$
(8.61)
$$\nabla^{2} p^{***}  \nabla^{2} p^{n} = \frac{{\rho^{n} }}{\Delta t}\nabla \cdot u^{**}$$
(8.62)

Step 4: First Corrector Step: Velocity

Step 5: Second Corrector Step: Pressure
$$\frac{{u^{n + 1}  u^{*} }}{\Delta t} = v\nabla^{2} u^{***}  \frac{1}{{\rho^{n} }}\nabla p^{****} + S^{u}$$
(8.63)
$$\nabla u^{****} = 0$$
(8.64)
$$\frac{{u^{****}  u^{***} }}{\Delta t} =  \frac{1}{{\rho^{n} }}\left( {\nabla p^{****}  \nabla p^{***} } \right) + \left( {v\nabla^{2} u^{***} } \right)  \left( {v\nabla^{2} u^{**} } \right)$$
(8.65)
$$\nabla^{2} p^{****}  \nabla^{2} p^{***} = \rho^{n} \nabla \cdot \left[ {\left( {v\nabla^{2} u^{***} } \right)  \left( {v\nabla^{2} u^{**} } \right)} \right]$$
(8.66)

Step 6: Second Corrector Step: Velocity

Step 7: SemiLagrangian Advection: Temperature
$$\frac{{T^{*}  T^{n} }}{\Delta t} =  \left( {u^{n} \cdot \nabla } \right)T^{*} \Rightarrow T^{*} = T^{n} \left[ {P\left( {x,  \Delta t} \right)} \right]$$
(8.67)

Step 8: Corrector Step: Temperature
$$\frac{{T^{n + 1}  T^{*} }}{\Delta t} = a\nabla^{2} T^{n + 1} + S^{T}$$
(8.68)

Step 9: Update of Density
$$\rho^{n + 1} = \frac{{p^{n + 1} M}}{{RT^{n + 1} }}$$
(8.69)
(3)
Simulation Cases
×
×
Simulation Cases

Results and Analysis:
(1)
Lid
driven cavity flow
×
Lid
driven cavity flow
(2)
2
D mixing convection flow
×
2
D mixing convection flow
When the study increases the grid number from 80 × 80 to 300 × 300, and further to 1000 × 1000, the computational cost performance for these algorithms changes as shown in Fig.
8.10a, b. As the number of grid increases, SLPISO and FFD are faster than PISO. The reason for this is the inherent characteristic of the semiLagrangian scheme. As the grid number increases, the computing cost of the traditional solvers, such as SIMPLE and PISO, demonstrates exponential growth trend, while the semiLagrangian scheme shows a linear growth as revealed in Fig.
8.10c (the influence of correction steps makes the calculation cost growth of FFD and SLPISO not exactly the linear).
×
The comparison of simulation speed above is under the situation of using the same time step. However, the stability analysis shows that SLPISO can tolerate a larger time step than PISO. The study uses the mixing convection case with mesh size of 1000 × 1000 to check the actual calculation speed of different solvers with different time steps. Figure
8.11 shows the computing time with the largest time step that each solver can handle. To reach stable and acceptable results for this case, the largest time steps are, 0.02 s, 0.005 s, 0.08 s, and 0.1 s, for SIMPLE, PISO, SLPISO and FFD, respectively. The shadowed columns in Fig.
8.11 show the relative computing cost with the time step size of 0.005 s for all the solvers, using SIMPLE as the benchmark. The black columns show the relative computing cost using their own largest time step. While the predicted results for velocity and temperature are similar to Fig.
8.9a, b, the modeling speeds of SLPISO and FFD with larger time steps are significantly increased.
×
(3)
Transient 2
D mixing convection flow
×
Transient 2
D mixing convection flow
The increasing deviation of the SLPISO results is attributed to the false diffusion of the time term. Compared to the original equation, the discretization of the time term leads to an additional false diffusion
\(\frac{{u^{2} \varDelta t}}{2}\nabla^{2} u\) that is related to the time step size. The increase of the time step enlarges the false diffusion, so that the transient simulation result is less responsive than the reference curve, to the transient velocity. If the constant effective kinematic viscosity is adjusted according lower, compensating for the larger time step used, the results of SLPISO with the time step of 0.08 s can be similar to the results with the time step of 0.005 s, as verified by the numerical tests.
(4)
Discussions

Objectives:
×
Objectives:
Key learning points:

Indoor airflow and heat transfer simulation

Simplification of indoor object (person)

Comparison of simulation with experimental data.

Case Descriptions:
(1)
3D computational domain with dimensions of X × Y × Z = 2.44 × 1.2 × 2.46 m.
(2)
Air is supplied through the full crosssectional area at one end of the channel and leaves through two circular openings at the opposite end.
(3)
The circular exhaust openings have a diameter of 0.25 m and are located 0.6 m from the floor and the ceiling, respectively.
(4)
The CSPs are located 0.7 m from the inlet, centered on the xaxis.
(5)
The geometry of the CSP is based on an averagesized woman with a standing height of 1.7 m. When seated, the CSP has a height of 1.38 m. The surface area of the CSP is 1.52 m
^{2}.
Pick up a reasonable body size.
(6)
A uniform velocity profile of U = 0.2 m/s and T = 22 °C is applied to the opening. Inlet turbulence intensity k and dissipation rate ε values can be calculated based on the literature (ISSN 13957953 R0307).
(7)
A convective heat flow rate of 38.0 W is prescribed for the CSP corresponding to an activity level of approximately 1 Met (sedentary work).
(8)
Steady state w/o contaminant.
(9)
Other surfaces are adiabatic.
(10)
More case details and
experimental data can be found at:
http://homes.civil.aau.dk/pvn/cfdbenchmarks/csp_benchmark_test/.

Simulation Details:
(2)
Convergence criterion: 0.1%.
(3)
Iteration: at least 1000 steps.
(4)
Grid: local refined grid with different total grid numbers.

Cases to Be Simulated:
(1)
KERNG model with
at least three different orders of grid numbers (e.g., 30 × 15 × 30, 45 × 23 × 45, 70 × 35 × 70).

Report:
(1)
Case descriptions: descriptions of the cases.
(2)
Simulation details: computational domain, grid cells, convergence status.

Figure of the best grid used (on X–Z and X–Y planes);

Figure of a typical convergence process recorded.
(3)
Result and analysis (only present the best results except for the 1st item).

Gridindependent solution: use one vertical pole at X = 1.69 m to compare and show the predicted velocity differences with different grids;

Figure of velocity contours at the middle height of the CSP;

Figure of airflow vectors at the middle height of the CSP;

Figure of temperature contours at the middle height of the CSP;

Figure of velocity contours at the central plane cross the CSP;

Figure of airflow vectors at the central plane cross the CSP;

Figure of temperature contours at the central plane cross the CSP;

Comparison of velocities along the three tested vertical poles at the central plane cross the CSP (experimentdot; simulationsolid line).
(4)
Conclusions (findings, CFD experience and lessons, etc.)

Figure of the best grid used (on X–Z and X–Y planes);

Figure of a typical convergence process recorded.

Gridindependent solution: use one vertical pole at X = 1.69 m to compare and show the predicted velocity differences with different grids;

Figure of velocity contours at the middle height of the CSP;

Figure of airflow vectors at the middle height of the CSP;

Figure of temperature contours at the middle height of the CSP;

Figure of velocity contours at the central plane cross the CSP;

Figure of airflow vectors at the central plane cross the CSP;

Figure of temperature contours at the central plane cross the CSP;

Comparison of velocities along the three tested vertical poles at the central plane cross the CSP (experimentdot; simulationsolid line).
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 Titel
 Solve Case
 DOI
 https://doi.org/10.1007/9789813298200_8
 Autor:

Zhiqiang (John) Zhai
 Verlag
 Springer Singapore
 Sequenznummer
 8
 Kapitelnummer
 Chapter 8