1、第11講,高精度有限差分方法,1,守恒型有限差分格式的構(gòu)造，有限差分WENO格式,2,守恒型有限差分格式的構(gòu)造,基本方程 均勻網(wǎng)格 已知某個(gè)時(shí)刻的 求數(shù)值通量函數(shù) 使得 半離散格式 +時(shí)間推進(jìn),3,構(gòu)造方法,直接對(duì)導(dǎo)數(shù)進(jìn)行離散，然后導(dǎo)出通量表達(dá)式 直接推導(dǎo)數(shù)值通量！,4,思路：如果找到一個(gè)函數(shù) ，滿足 因此，只需使下面關(guān)系成立， 則有,5,則顯然有,如何得到 定義原函數(shù) 則 比較有限體積方法: : 以 為中心的 寬度區(qū)間內(nèi) 的平均值,6,已知 ，則可以確定 對(duì) 做Lagrange插值（先要選定模板） 可得： 求導(dǎo)： ENO/WENO格式的界面處物理量值的計(jì)算方法適用于計(jì)算有限差分方法數(shù)值通量！

2、,7,注意和有限體積方法的區(qū)別！,8,在實(shí)際實(shí)施過程中，感興趣的是控制體界面處的左右狀態(tài)。我們利用 計(jì)算 Shu給出了具體公式如下：,均勻網(wǎng)格,9,具體公式（均勻網(wǎng)格）,10,基于ENO/WENO的有限差分格式,標(biāo)量方程 均勻網(wǎng)格,半離散格式 通量,11,時(shí)間積分 ENO/WENO重構(gòu)的目的是計(jì)算正負(fù)通量,12,ENO/WENO重構(gòu),格式的空間精度取決于ENO或WENO重構(gòu)的階數(shù)。,13,基于ENO/WENO的有限差分格式,方程組 矢通量分裂 采用StegerWarming或Van Leer的方法把通量分解成 用ENO（WENO）插值方法計(jì)算單元界面正負(fù)通量 計(jì)算通量函數(shù) 選擇合適的時(shí)間積分方

3、法(如RungeKutta方法) 推進(jìn)求解,14,每個(gè)分量分別重構(gòu)!,特征分解 對(duì)于固定的 1 2確定i和i1點(diǎn)ENO/WENO重構(gòu)需要用到的所有可能的模板對(duì)應(yīng)的網(wǎng)格點(diǎn), 在這些網(wǎng)格點(diǎn)計(jì)算 3對(duì)于特征通量的每個(gè)分量進(jìn)行ENO重構(gòu), 計(jì)算 4,15,緊致格式,16,常規(guī)差分近似,導(dǎo)數(shù)的差分近似1：常規(guī)方法 待定參數(shù) 最高精度 中有一個(gè)參數(shù)可取任意值 中有兩個(gè)參數(shù)可取任意值 依次類推,17,緊致差分近似,導(dǎo)數(shù)的差分近似2：緊致格式(Lele JCP,1992) 系數(shù)確定：兩側(cè)分別作Taylor展開 待定系數(shù)：k+l+m+n+2；最高精度：k+l+m+n+1 的確定：求解方程組，需邊界條件、邊界格式

4、,18,守恒型緊致差分格式的構(gòu)造,基本方程 均勻網(wǎng)格 已知某個(gè)時(shí)刻的 求數(shù)值通量函數(shù) 使得 半離散格式 +時(shí)間推進(jìn),19,思路：如果找到一個(gè)函數(shù) ，滿足 因此，只需使下面關(guān)系成立， 則有,20,則顯然有,如何得到 定義原函數(shù) 則 比較有限體積方法: : 以 + 1 2 為中心的 寬度區(qū)間內(nèi) 的平均值,21,已知 ，則可以確定 利用 用緊致格式計(jì)算 兩種方法,22,1,2,確定系數(shù) 實(shí)際計(jì)算,守恒型迎風(fēng)緊致格式,守恒型左偏心緊致格式,23,，,，,階格式,守恒型迎風(fēng)緊致格式,守恒型右偏心緊致格式,24,，,，,階格式,，,，,最終格式,25,緊致格式的問題,要求在差分涉及的模板上解是光滑的。 不

5、能計(jì)算有間斷的流動(dòng) 在間斷附近有數(shù)值振蕩 可能導(dǎo)致計(jì)算發(fā)散 解決方法 緊致WENO混合格式,26,Hybrid Compact-WENO Scheme: Scalar Case (1),Hybrid Compact-WENO Scheme: Scalar Case (2),Hybrid Compact-WENO Scheme: Scalar Case (3),Hybrid Compact-WENO Scheme: Scalar Case (4),Hybrid Compact-WENO Scheme: Euler Equations (1),Eigenvalues and Eigenvector

6、s at Local characteristic variables Hybrid scheme in terms of characteristic variables,Hybrid Compact-WENO Scheme: Euler Equations (2),Final Scheme,Test Cases(1),Shu-Osher Problem, WENO,Test Cases(2),Shu-Osher Problem,Test Cases(3),Shu-Osher Problem,Test Cases(4),Shu-Osher Problem,Test Cases(5),Doub

7、le Mach Reflection,Test Cases(6),Double Mach Reflection: Present Method,Test Cases(7),Double Mach Reflection: WENO Method,Test Cases(8),Double Mach Reflection: Pirozzoli Method,Test Cases(9),Shock-Vortex Interaction,Test Cases(10),Shock-Vortex Interaction: Present Method,Test Cases(11),Shock-Vortex

8、Interaction: WENO Method,Test Cases(12),Shock-Vortex Interaction: Pirozzoli Method,差分格式的色散耗散特性及其優(yōu)化,45,46,1. 差分格式的色散與耗散,46,色散和耗散 是周期為L(zhǎng)的周期性函數(shù)，把0，L上劃分為N等分，網(wǎng)格間距h=L/N，對(duì) 作Fourier級(jí)數(shù)展開,47,無量綱化,導(dǎo)數(shù)精確值,色散和耗散 導(dǎo)數(shù)的差分近似,48,逼近,的程度代表了差分近似的色散和耗散,色散和耗散,49,耗散,色散,50,a 二階中心差分 b 四階中心差分 c 四階中心型緊致格式 d 六階中心型緊致格式,色散關(guān)系,51,耗散關(guān)系

9、,52,色散和耗散的優(yōu)化準(zhǔn)則： Dispersion: should be as small as possible; Dissipation: A small amount of dissipation is necessary Central difference scheme may be insufficient in suppressing the numerical oscillation and can lead to instability: (Lechner, 2001) In the range of high wave numbers, the waves propaga

10、te at an incorrect speed, and it can be desirable to damp them as much as possible (Pirozzoli, 2002) The optimal value of dissipation is problem-dependent The bandwidth-optimized WENO scheme (Martin, 2006) works well in the DNS of supersonic boundary layer, but causes oscillations in the case studie

11、d by Cai (2008) It is desirable for a scheme to have minimized dispersion and controllable dissipation（MDCD）.,52,53,2.線性色散最小、耗散可控差分格式(MDCD),53,54,MDCD -FD scheme,Finite difference discretization of using (2r+1) symmetrical stencils,Lemma 1：If approximates to （2r2）th order of accuracy on (2r+1) symme

12、trical stencils, the dispersion and dissipation of the corresponding semi-discrete scheme are independent of each other.,54,55,MDCD -FD scheme,If approximates to (2r -2)th order of accuracy, it can be written in the following general form,where,55,56,MDCD -FD scheme,Fourier transformation：,Dispersio

13、n,Dissipation,56,57,MDCD -FD scheme,Lemma 2：If with 2n free parameters approximates to (2r-2n)th order of accuracy on (2r+1) symmetrical stencils, then the dispersion of the corresponding semi-discrete scheme is determined by n free parameters and the dissipation is determined by other n free parame

14、ters. Lemma 1 and Lemma 2 make it possible to develop finite difference schemes with minimized dispersion and controllable dissipation.,57,58,Dispersion and Dissipation of MDCD,For the case r=3 （Using Lemma 1 ）,Dispersion :,Dissipation :,58,59,The optimal value of is obtained by the minimization of

15、the following function ：,The dispersion properties of the scheme is determined by,The dissipation properties of the scheme is determined by,Dispersion and Dissipation of MDCD,59,60,Dispersion and Dissipation,Compare of the dispersion properties,60,優(yōu)化結(jié)果：=8 disp=0.046378,61,Dispersion and Dissipation,

16、The dissipation of MDCD can range from zero to an arbitrary value The dissipation of UW5 and C6 is fixed,為保證格式穩(wěn)定性， 通常取,62,63,3. MDCD-WENO混合格式,63,64,Hybrid scheme: MDCD-HY,rc=0.4,64,可進(jìn)一步推廣到方程組、多維問題，NS方程，,65,4. 計(jì)算結(jié)果,65,66,Linear wave equation：,Linear transport equation,Initial condition：,A wave packet

17、 characterized by sine shape wave with different frequency. As m becomes larger, it contains more high wave number elements,MDCD-HY(0),66,67,Linear transport equation,67,68,Linear transport equation,68,69,Shu-Osher problem：,This test case represents a Mach 3 interacting a sine entropy wave. Both sma

18、ll scale features and discontinuities are contained.,Initial condition：,Computational condition： grid points: N=200 CFL=0.2,MDCD-HY(0),MDCD-HY(0.015),69,70,Shu-Osher problem：,70,71,Propagation of broadband sound wave,A sound wave packet which contains acoustic turbulence Characterized by different l

19、ength scale and a given spectrum.,Initial condition：,Energy spectrum：,Computational condition： grid points: N=128 CFL=0.2,MDCD-HY(0),71,72,Propagation of broadband sound wave,Broadband sound wave propagation k0=4,72,73,Propagation of broadband sound wave,Broadband sound wave propagation k0=8,73,74,P

20、ropagation of broadband sound wave,Broadband sound wave propagation k0=12,74,75,2-D viscous shock tube,Computational condition： Grid points: N=300*150 CFL=0.3,Rehaman et al (AIAA, 2010),75,76,2-D viscous shock tube,Case A Contour of density,76,77,2-D viscous shock tube,Case B Contour of density,77,7

21、8,2-D viscous shock tube,Case C Contour of density,MDCD-HY(0.046),78,79,2-D viscous shock tube,Comparison of CPU time,MDCD-HY is about 15% more efficient than WENO-JS,79,X：streamwise,Y：spanwise,Z：wall-normal,Relevant parameters for DNS,DNS of compressible wall turbulence,Boundary condition： Periodic

22、 in streamwise and spanwise direction Wall: no-slip boundary condition, isothermal wall,80,Case A: (Mam=0.35) ： One-dimensional energy spectral at Z+=20,DNS of compressible wall turbulence,81,Case A: (Mam=0.35): Two-point correlations at Z+=20：,DNS of compressible wall turbulence,82,Case A: (Mam=0.3

23、5) statistic quantities：,Mean streamwise velocity,RMS velocity,Reynolds stress,DNS of compressible wall turbulence,83,Case A: (Mam=0.35) ：turbulent flowfield,vortex structures,Streaks at z+=15,DNS of compressible wall turbulence,84,Case B: (Mam=1.5) statistic quantities：,Mean density, velocity and t

24、emperature,RMS velocity,RMS temperature,RMS density,DNS of compressible wall turbulence,85,Case B: (Mam=1.5) One-dimension velocity energy spectra with different grid points ：,DNS of compressible wall turbulence,Streamwise velocity,Spanwise velocity,Wall-normal velocity,86,Turbulent flow in a channe

25、l with wavy wall,Computational domain: 8*2*4 (h=2),Geometry：,Reynolds number：,87,(Mam=0.35) Turbulent statistic at crest and the valley ：,streamwise velocity,Wall-normal velocity,RMS streamwise velocity,RMS wall-normal velocity,Turbulent flow in a channel with wavy wall,88,(Mam=0.35) turbulent flowf

26、ield：,Near wall vortex structures,Turbulent flow in a channel with wavy wall,89,Smart wall Opposition velocity control:,We restrict the maximum amplitude of wall deformation to,Flow velocity at,Active control: Flow control using the smart wall approach,90,The reduction of skin friction is nearly 40%

27、 The reduction of total drag is about 30%,Case A: (Mam=0.35),Active control: Flow control using the smart wall approach,91,The reduction of skin friction is about 20% The pressure drag is negligible,Case B: (Mam=1.5),Active control: Flow control using the smart wall approach,92,Without Control,With

28、Control,Active control: Flow control using the smart wall approach,Contour of streamwise vorticity,93,Active control: Flow control using the smart wall approach,Without Control,With Control,Near wall vortex structures,94,95,Flow control using the smart wall approach,Wall deformation and near wall flow structures,ParCFD2010,95,Active dimples：,Control scheme：,Number of dimples：3232,Active control: active dimple,Longer in streamwise direction,96,Active control: active dimple,Reduction ra