Patankar

Cheatsheet Content

1. Introduction to FVM Finite Volume Method (FVM): Discretizes the governing equations (PDEs) into a set of algebraic equations for control volumes. Conservation: FVM inherently conserves quantities (mass, momentum, energy) over each control volume. Control Volume: A small, non-overlapping region of the domain where the conservation equations are applied. 2. General Transport Equation The steady-state general transport equation for a scalar $\phi$ can be written as: $$ \frac{\partial}{\partial x}(\rho u \phi) + \frac{\partial}{\partial y}(\rho v \phi) = \frac{\partial}{\partial x}\left(\Gamma \frac{\partial \phi}{\partial x}\right) + \frac{\partial}{\partial y}\left(\Gamma \frac{\partial \phi}{\partial y}\right) + S_\phi $$ $\rho$: Density $u, v$: Velocity components $\phi$: Scalar variable (e.g., temperature, concentration) $\Gamma$: Diffusion coefficient $S_\phi$: Source term 3. Discretization of 1D Convection-Diffusion Equation Consider the 1D steady-state convection-diffusion equation: $$ \frac{d}{dx}(\rho u \phi) = \frac{d}{dx}\left(\Gamma \frac{d\phi}{dx}\right) + S_\phi $$ Integrating over a control volume $P$ from $x_w$ to $x_e$: $$ (\rho u \phi)_e - (\rho u \phi)_w = \left(\Gamma \frac{d\phi}{dx}\right)_e - \left(\Gamma \frac{d\phi}{dx}\right)_w + \bar{S}_\phi \Delta x $$ $e, w$: East and West faces of the control volume. $\Delta x$: Width of the control volume. $\bar{S}_\phi$: Average source term over the control volume. 4. Approximation Schemes for Face Values a. Central Differencing Scheme (CDS) Approximation: $\phi_e \approx \frac{\phi_P + \phi_E}{2}$, $\phi_w \approx \frac{\phi_W + \phi_P}{2}$ Diffusion term: $\left(\Gamma \frac{d\phi}{dx}\right)_e \approx \Gamma_e \frac{\phi_E - \phi_P}{\delta x_{PE}}$ Convection term: Uses linear interpolation. Issue: Can lead to oscillations for high Peclet numbers ($Pe > 2$). b. Upwind Differencing Scheme (UDS) Approximation: $\phi_e = \begin{cases} \phi_P & \text{if } (\rho u)_e > 0 \\ \phi_E & \text{if } (\rho u)_e Advantages: Always stable, oscillation-free. Disadvantages: Highly diffusive, introduces numerical error (false diffusion). c. Power-Law Scheme (Patankar's Scheme) Motivation: Aims to combine stability of UDS with accuracy of CDS, especially for convection-dominated flows. Peclet Number: $Pe = \frac{\rho u \Delta x}{\Gamma}$ (ratio of convection to diffusion). Idea: For $Pe \leq 2$, use CDS. For $Pe > 2$, switch to UDS. Approximation Function $A(|Pe|)$: $$ A(|Pe|) = \max[0, (1 - 0.1|Pe|)^5] $$ This function is used to blend diffusion and convection contributions. Face Fluxes (1D Example): Consider the flux $F_e = (\rho u)_e$ and $D_e = \Gamma_e / \delta x_{PE}$. The net flux through face $e$ is $F_e \phi_e - D_e (\phi_E - \phi_P)$. The Power-Law scheme approximates the combined convection-diffusion coefficient. 5. Patankar's Power-Law Discretization Coefficients (1D) For a control volume $P$ with neighbors $W$ and $E$, the discretized equation is: $$ a_P \phi_P = a_W \phi_W + a_E \phi_E + S_P $$ Where: $F_w = (\rho u)_w$, $F_e = (\rho u)_e$ $D_w = \Gamma_w / \delta x_{WP}$, $D_e = \Gamma_e / \delta x_{PE}$ $a_W = D_w A(|Pe_w|) + \max(F_w, 0)$ where $A(|Pe_w|) = \max[0, (1 - 0.1|Pe_w|)^5]$ and $Pe_w = F_w/D_w$. $a_E = D_e A(|Pe_e|) + \max(-F_e, 0)$ where $A(|Pe_e|) = \max[0, (1 - 0.1|Pe_e|)^5]$ and $Pe_e = F_e/D_e$. $a_P = a_W + a_E + (F_e - F_w) - S_P$ (where $S_P$ is part of the linearized source term $S_\phi = S_C + S_P \phi_P$) The term $(F_e - F_w)$ accounts for the net flow rate through the control volume. For source term linearization: $S_\phi = S_C + S_P \phi_P$. Here $S_P$ is usually negative to ensure stability. 6. Properties of Power-Law Scheme Boundedness: Ensures that the solution remains within physical bounds (e.g., $\phi_{min} \leq \phi \leq \phi_{max}$). This is achieved by the coefficients $a_W, a_E, a_P$ satisfying certain conditions. Monotonicity: Produces oscillation-free solutions. Accuracy: More accurate than UDS, especially at higher Peclet numbers, by incorporating diffusion for $Pe \leq 2$. Robustness: Performs well over a wide range of Peclet numbers. 7. Extension to 2D/3D The 1D approach extends to 2D/3D by considering fluxes through all faces of the control volume (North, South, East, West for 2D). The general discretized equation becomes: $$ a_P \phi_P = a_W \phi_W + a_E \phi_E + a_N \phi_N + a_S \phi_S + S_C $$ The coefficients $a_N, a_S$ are derived similarly to $a_W, a_E$ considering the $y$-direction convection-diffusion terms. 8. Staggered Grid for Velocity Issue: If velocities are stored at cell centers, pressure and velocity fields can become decoupled, leading to checkerboard pressure fields. Solution: Staggered grid where velocity components are stored at control volume faces (e.g., $u$ velocity at east/west faces, $v$ velocity at north/south faces). 9. SIMPLE Algorithm (Semi-Implicit Method for Pressure-Linked Equations) Purpose: To solve pressure-velocity coupling in incompressible flows. Steps: Guess pressure field $P^*$. Solve momentum equations using $P^*$ to get $u^*, v^*$. Solve pressure correction equation for $P'$. Update pressure: $P = P^* + P'$. Update velocities: $u = u^* + u'$, $v = v^* + v'$. Repeat until convergence. Pressure Correction Equation: Derived from the continuity equation and relates $P'$ to the continuity imbalance.