Solving the Navier-Stokes Equations

Transcription

1 FMIA F. Moukalled L. Mangani M. Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical oundation o the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation o the FVM numerics and algorithms used in the simulation o incompressible and compressible luid lows, along with a detailed examination o the components needed or the development o a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The irst is ufvm, a three-dimensional unstructured pressure-based inite volume academic CFD code, implemented within Matlab. The second is OpenFOAM, an open source ramework used in the development o a range o CFD programs or the simulation o industrial scale low problems. Moukalled Mangani Darwish Fluid Mechanics and Its Applications 3 Series Editor: A. Thess The Finite Volume Method in Computational Fluid Dynamics With over 220 igures, numerous examples and more than one hundred exercises on FVM numerics, programming, and applications, this textbook is suitable or use in an introductory course on the FVM, in an advanced course on CFD algorithms, and as a reerence or CFD programmers and researchers. Fluid Mechanics and Its Applications F. Moukalled L. Mangani M. Darwish The Finite Volume Method in Computational Fluid Dynamics The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction with OpenFOAM and Matlab Engineering ISBN Solving the Navier-Stokes Equations Chapter 6

2 Pressure Equation or Compressible Flow

3 Compressible Flow ρ t + ρv = 0 ( ρv) + ( ρvv) = τ p + B t ρ = C ρ p ρ = ρ ( P + P ) = ρ ( P) + ρ P p ρ P p = C ρ p p = p (n) + p ρ = ρ (n) + ρ v = v * +

4 Discretized Equations ρ t + ρv = 0 Incompressible ( m + m ) = 0 = nb(p ) Compressible (n) ( ρ P + ρ P ρ P ) V P + (!m +!m ) = 0 =nb(p)!m = ρ ( v + ) S = ρ v S " $ # %$ + ρ v " $ # S %$!m!m v +!m = ρ (n) + ρ S = ρ (n) v S " $ # %$ + ρ (n) S + ρ v S + ρ S " $$$$ $ # $$$$$$ %!m!m

5 Velocity Correction ρ (n) v S " $ # %$ + ρ (n) S + ρ v S + ρ S " $$$$ $ # $$$$$$ %!m! m!m = ρ (n) v S ρ (n) D ( p (n) (n) p ) S (n) m! = ρ S ρ (n) D ( p p ) S " $$$$$$ # $$$$$$ % +!m ρ S (n) C ρ, p () " $ $ # $$$ % (2)

6 Pressure Equation (n) ( ρ C + ρ C ρ C ) + (!m +!m ) = 0 =nb(c ) V P C!m ρ p C + ρ (n) D p S + (n) ρ C p ρ = =nb(p) ρ (n) o C ρ C + =nb(c )!m S ρ (n) v =nb(c ) =nb(c ) ρ S ( ρ + D p ) S = H [ ] S = 0.5 ( H C + H N ) S =nb(c ) =nb(p) =nb(c ) = 0.5 =nb(c ) a NBP NBP a P + NBP(C ) NB(F ) a NBF a F v NBF S

7 Pressure Equation C ρ!# " $# p C transient like term %m + C ρ (n) ρ p + ρ (n) D ( p ) S =nb(c ) =nb(c )!### "### $!####" #### $ convection like term diusion like term ρ (n) & C ρ C = + %m =nb(c )!##### "##### $ source like term =nb(p) a C = C ρ + ρ (n) D p C ρ,!m (n) ρ,0 + ρ (n) D =nb(c ) =nb(c ) S =!m C ρ (n) ρ p =!m,0 C ρ, =nb(c ) =nb(p) ρ (n) D p p (n) C!m,0 C ρ, (n) ρ ρ E + T p F = ρ (n) D p F p C =nb(p) D = d u E x, + d v E y, d PF a F =!m,0 C ρ, ρ (n) (n) D ρ b C = ρ (n) " ( C ρ C ) + =nb(c )!m + =nb(c ) ρ (n) ( D p ) T

8 Pressure Equation C ρ ( p C )+ C ρ U p + ρ D ( p p )!###" ### $ Rhie-Chow interpolation S = ρ % ( C ρ C ) &m ρ S =nb( C) =nb( C) C ρ ( p C )+ C ρ U P =nb( C) High Resolution ρ D ( p ) S =nb( C) = ρ! ( C ρ C ) ρ * * ( U ) ρ v S ρ + D P =nb C =nb C =nb C Neglect S C ρ ( p C!m )+ C ρ =nb( C) ρ p a C p C + a F F=NB(C ) ( ρ D ( p ) S ) =nb( C) p F = b C = Ω ρ o ( P ρ P )+!m =nb( C) " $$$$ # $$$$ % p,v,!m Residual SIMPLE SIMPLEC SIMPLER SIMPLEST SIMPLE-M PISO treatment leads to variety o schemes

9 All-Speed Flow Algorithms Transient-like Term ΩC ρ * ( P P ) + ( C ρ U P ) ( ρ * D ( P ) S ) Advection-like Term account or compressibility eects = Ω ρ * o P ρ P + ρ * * ( U ) ( ρ v S ) ρ * H v ( [ ] S )

10 Multigrid Acceleration M. DARWISH ET AL M. DARWISH ET AL Figure 6. (a) Pressure contours and (b) Mach number distributions along the upper and lower walls or transonic low over a bump (Minlet ¼ 0.675). Convergence history plots o the various algorithms using the (c) single-grid, (d ) prolongation grid, and (e) multigrid methodologies or transonic low over a bump (Minlet ¼ 0.675). Figure 7. (a) Pressure contours and (b) Mach number distributions along the upper and lower walls or supersonic lo history plots o the various algorithms using the (c) single-grid, (d ) prolongation grid, and (e) multigrid methodologies o ersonic low over a bump (Minlet ¼.4). Convergence ologies or supersonic low over a bump (Minlet ¼.4). Figure 6. (a) Pressure contours and (b) Mach number distributions along the upper and lower walls or transonic low over a bump (Minlet ¼ 0.675). Conver Figure (a) (e) Pressure contours and (b) Mach number distributions along (M the upper history plots o the various algorithms using the (c) single-grid, (d ) prolongation grid,7.and multigrid methodologies or transonic low over a bump inlet ¼ 0 history plots o the various algorithms using the (c) single-grid, (d ) prolongation grid,

11 Problem - Staggered Grid Use the SIMPLE procedure to compute p2, ub, and uc rom the ollowing data: As an initial guess, set u B uc 2 3