Presentation on weno lbfs
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This document presents a yearly project report on using the Lattice Boltzmann Method (LBM) to model supersonic and hypersonic flows. It discusses developing a hybrid Finite Difference Weighted Essentially Non-Oscillatory (FD-WENO) and Lattice Boltzmann Flux Solver (LBFS) scheme in MATLAB. Several test cases are modeled and compared to exact solutions, with some deviations found. Future work includes improving the code, extending it to 2D, implementing it in C++, and developing a Double Distribution Function LBM model. Expenditure details of 3,56,499/- for a computer workstation are also provided.
Presentation on weno lbfs
- 1. A yearly project report On LATTICE BOLTZMANN METHOD FOR SUPERSONIC AND HYPERSONIC FLOWS by - Dr. Krunal M. Gangawane (Assistant Professor) & Varshan Sai Kumar Reddy Kadiveti (Research Fellow) Department of Chemical Engineering, National Institute of Technology (NIT) - Rourkela, Rourkela-769008 Tel.: +91 661 246 2253 Email: gangawanek@nitrkl.ac.in 1
- 2. CONTENTS • Introduction • FD - WENO (Finite difference Weighted non-oscillatory) Scheme • LBFS (Lattice Boltzmann flux solver) • Problems considered • Results and discussion • Future work plan Yearly report, LBM for supersonic and hypersonic flows 2
- 3. Introduction • Numerical schemes for compressible flows were enormously developed to provide approximate results with some allowable error. • For our research we are considering 1D inviscid compressible flow problem. • We are trying to understand existing numerical schemes and build a CFD code that can be utilized to capture shockwaves at high speed flows. • From the literature review we have found that the most challenging task is to capture contact discontinuity using numerical scheme due to presence of shock wave. Yearly report, LBM for supersonic and hypersonic flows 3
- 4. • Godunov proposed first discontinuous decomposition algorithm. This is only first- order accurate and it can not capture contact discontinuity effectively. • Van leer worked on the Godunov scheme and developed a monotonic upstream centered Scheme (MUSCL). It its second-order accurate. The numerical oscillations cannot be avoided by using this scheme. • The Godunov scheme is further developed by Colella, Woodward and made piecewise parabolic method (PPM). • This scheme can achieve third-order accuracy and the capturing of shock waves has been improved. Yearly report, LBM for supersonic and hypersonic flows 4
- 5. • All the previously described higher-order schemes have to use slope limiter to avoid numerical oscillations. It was a massive setback for these schemes. • To overcome this Harten has proposed an essentially non-oscillation (ENO) scheme. This scheme can achieve higher-order accuracy by avoiding numerical oscillations. • ENO scheme chooses three stencils among them only the smoothest stencil is considered for analysis by neglecting other two stencils. • The ENO scheme is further developed by Liu, Chan and Osher to make WENO scheme and finite volume (FV)-WENO scheme. • Later Jiang and Shu has proposed finite difference (FD) -WENO fifth order scheme. • Shu had found FD-WENO scheme is best to be consider for its accuracy, less complex calculations, and decent higher resolution. • In our project, we are considering the FD-WENO scheme for its advantages. Yearly report, LBM for supersonic and hypersonic flows 5
- 6. FD -WENO (Finite difference Weighted non-oscillatory) Scheme Above figure shows the stencils (S0,S1,S2) used in ENO and WENO scheme • Instead of using just one stencil as mentioned in ENO scheme, we are considering linear combination of all three stencils in WENO scheme. • This scheme has higher accuracy compared with all the schemes. Yearly report, LBM for supersonic and hypersonic flows 6
- 7. • WENO scheme is a polynomial reconstruction procedure that is adaptively switching from a high order polynomial to a nonlinear weighted sum of lower-order polynomials. • Nonlinear weights are designed based on the local smoothness indicators (ᵦ1, ᵦ2 , ᵦ3) of the lower order polynomials. • The success of WENO scheme depends on the Non-linear weights (w1,w2,w3). • FD-WENO scheme can achieve high resolution in smooth regions, and it can also capture discontinuities at the same time by neglecting local oscillations. Yearly report, LBM for supersonic and hypersonic flows 7
- 8. LBFS (Lattice Boltzmann flux solver) • Local Riemann solver is the efficient numerical scheme to solve compressible flow problems. • Ji proposed new flux solve called LBFS to over come problems faced by conventional flux solvers. • LBFS can only be used for inviscid compressible or incompressible flows. • In LBFS, the inviscid flux at the cell interface is reconstructed from the local solution of 1D compressible lattice Boltzmann model. • Here we are considering D1Q4 non- free parameter LBM model to analyze compressible flows for wide range of Mach number and for its faster convergence rate. Yearly report, LBM for supersonic and hypersonic flows 8
- 9. • In D1Q4 model, the non-equilibrium distribution function introduces the numerical dissipation for calculation of inviscid flows. • For solving viscous flows this numerical dissipation has to be controlled. Its under research. • Finite difference formulation for hyperbolic conservation laws requires high-order numerical fluxes to be applied at cell boundaries to form the flux differences across the spaced cells. • The conservative property of the spatial discretization is obtained by implicitly defining the numerical flux function. • After going through various literatures we have decided to work on FD (Finite difference)-WENO (Weighted essentially non-oscillation)- LBFS (Lattice Boltzmann flux solver) scheme Yearly report, LBM for supersonic and hypersonic flows 9
- 10. Problems considered For our analysis we have considered the following cases: 1) Sod shock tube problem 2) Right expansion and left strong shock 3) Double Expansion 4) Left Expansion and right strong shock 5) Lax test case 6) Mach=3 test case 7) Shock tube problem with supersonic zone Yearly report, LBM for supersonic and hypersonic flows 10
- 11. Results and Discussion • We have developed a WENO-LBFS hybrid solver in MATLAB. The high-speed compressible flow Matlab code has been developed for 1D by taking into account the following points. • The parameters are reconstructed using the 5th order WENO scheme. • Numerical fluxes are evaluated using D1Q4 non-free parameter model. • To update conservation variables (U), a three-step Runge-Kutta scheme is used. • Shu has suggested taking CFL ≤ 0.5 to maintain essentially non-oscillatory nature of the WENO scheme. • In our analysis, we are considering the CFL value as 0.25. • For different cases the WENO-LBFS scheme soluction and Exact solution has been compared. • The cases are shown in the following slides. Yearly report, LBM for supersonic and hypersonic flows 11
- 12. Case 1: Sod shock tube Problem Initial Conditions - Left Pressure (𝑝𝐿) = 1 Right Pressure (𝑝𝑅) = 0.1 CFL – 0.25 Left velocity (𝑢𝐿) = 0 Right velocity (𝑢𝑅) = 0 Max Time – 0.1 Left density (ρ𝐿) = 1 Right density(ρ𝑅) = 0.125
- 13. Case 2: Right expansion and left strong shock Initial Conditions - Left Pressure (𝑝𝐿) = 7 Right Pressure (𝑝𝑅) = 10 CFL – 0.25 Left velocity (𝑢𝐿) = 0 Right velocity (𝑢𝑅) = 0 Max Time – 0.1 Left density (ρ𝐿) = 1 Right density(ρ𝑅) = 1
- 14. Case 3: Double Expansion Initial Conditions - Left Pressure (𝑝𝐿) = 1 Right Pressure (𝑝𝑅) = 0.1 CFL – 0.25 Left velocity (𝑢𝐿) = 0 Right velocity (𝑢𝑅) = 0 Max Time – 0.03 Left density (ρ𝐿) = 1 Right density(ρ𝑅) = 0.125
- 15. Case 4: Left Expansion and right strong shock (Shock tube problem) Initial Conditions - Left Pressure (𝑝𝐿) = 1 Right Pressure (𝑝𝑅) = 0.1 CFL – 0.25 Left velocity (𝑢𝐿) = 0.75 Right velocity (𝑢𝑅) = 0 Max Time – 0.17 Left density (ρ𝐿) = 1 Right density(ρ𝑅) = 0.125
- 16. Case 5: Lax test case Initial Conditions - Left Pressure (𝑝𝐿) = 3.528 Right Pressure (𝑝𝑅) = 0.571 CFL – 0.25 Left velocity (𝑢𝐿) = 0.698 Right velocity (𝑢𝑅) = 0 Max Time – 0.15 Left density (ρ𝐿) =0.445 Right density(ρ𝑅) = 0.5
- 17. Case 6: Mach = 3 test case Initial Conditions - Left Pressure (𝑝𝐿) = 10.333 Right Pressure (𝑝𝑅) = 1 CFL – 0.25 Left velocity (𝑢𝐿) = 0.92 Right velocity (𝑢𝑅) = 3.55 Max Time – 0.09 Left density (ρ𝐿) =3.857 Right density(ρ𝑅) = 1
- 18. Case 7: Shock tube problem with supersonic zone Initial Conditions - Left Pressure (𝑝𝐿) = 1 Right Pressure (𝑝𝑅) = 0.02 CFL – 0.25 Left velocity (𝑢𝐿) = 0 Right velocity (𝑢𝑅) = 0 Max Time – 0.16 Left density (ρ𝐿) =1 Right density(ρ𝑅) = 0.02
- 19. Future work plan • In the above cases, few FD WENO LBM scheme graphs deviate from the exact solution. • We will be further working on our code to develop it better for all cases. Few more cases will be included. • FD-WENO LBFS developed above is in one dimension. It will be further developed into two dimensions and in C++ programming language. • Development of the double distribution function (DDF) LB code based upon the model of Li et al. (2010) is UNDER PROCESS. • We will be working on the code template given by ISRO.
- 20. Expenditure details • Equipment : Computer workstation (Received 3 tendors, L1: 3,56,499/-) Yearly report, LBM for supersonic and hypersonic flows 20