Unified methods for computing incompressible and compressible flow

Computation of incompressible and compressible flow generally depends on the Mach number M, where for a range of zero to supersonic compressible equations are applied but with a possible error on a range of M<0.2. For this range we have to apply incompressible Navier Stokes and Euler equations but the work would be much easier if we find a Unified Method for solving both the flows. Unified method can also lead us towards much more accuracy and efficiency.

The standard method for solving compressible flows fails; the basic cause of failure for the compressible flow methods is the stiffness of the governing equations.

Conservation of mass

${\displaystyle {\frac {\partial {}\rho {}}{\partial {}t}}+\nabla {}.\left(\rho {}V\right)=0}$

Conservation of momentum

${\displaystyle \rho {}{\frac {DV}{Dt}}=-\nabla {}p+\nabla {}.\tau {}+\rho {}f}$

Conservation of energy

${\displaystyle \rho {}\left[{\frac {\partial {}h}{\partial {}t}}+\nabla {}.\left(hV\right)\right]=-{\frac {Dp}{Dt}}+\nabla {}.\left(k\nabla {}T\right)+\Phi {}}$

One way to fix this problem is to change the governing equation; known as preconditioning; which can also increases the accuracy.

The other cause for the breakdown is pressure because it is not taken into account as primary unknown. For making the governing equation workable for both the compressible and incompressible flows, following things needs to be corrected:-

• Usage of dimensionless pressure thereby removing the difficulties faced while solving for very low Mach number
• Use non conservative form of energy which increases the efficiency
• Discretization of the mass conservation equation
• Use MUSCL and Runge–Kutta time stepping

Governing equation

Conservation of mass

${\displaystyle {\rho {}}_{p}p_{t}+{\rho {}}_{T}T_{t}+m_{x}=0}$

Equation of state

${\displaystyle {\rho {}}_{p}=\gamma {}M_{r}^{2}/T}$

${\displaystyle {\rho {}}_{T}=-{\frac {\rho {}}{T}}}$

Momentum equation

${\displaystyle m_{t}+{(um+p)}_{x}=0}$

By using the dimensionless pressure and equation of state the governing equation can be best described as: ${\displaystyle T_{t}+{(uT)}_{x}+\left(\gamma {}-2\right)Tu_{x}=0}$

Finite volume scheme

For the above specified governing equations the finite volume scheme ^[1] is

${\displaystyle \gamma {}M_{r}^{2}\left(p_{j}^{n+1}-p_{j}^{n}\right)-{\rho {}}_{j}^{n}\left(T_{j}^{n+1}-T_{j}^{n}\right)+\lambda {}T_{j}^{n}\left(s^{n}{\rho {}}^{n}u^{n}+\left(1-s^{n}\right)m^{n+1}\right){\vert {}}_{j-1/2}^{j+1/2}=0}$

${\displaystyle m_{j+1/2}^{n+1}-m_{j+1/2}^{n}+\lambda {}\left(u^{n}m^{n}+p^{n+1/2}\right){\vert {}}_{j}^{j+1}=0}$

${\displaystyle T_{j}^{n+1}-T_{j}^{n}+\lambda {}\left(u^{n}T^{n}\right){\vert {}}_{j-{\frac {1}{2}}}^{j+{\frac {1}{2}}}+\lambda {}\left(\gamma {}-2\right)T_{j}^{n}u^{n}{\vert {}}_{j-{\frac {1}{2}}}^{j+{\frac {1}{2}}}=0}$

where ${\displaystyle \lambda {}=\tau {}/h}$

${\displaystyle p^{n+1/2}=(p^{n}+p^{n+1})/2}$

${\displaystyle s^{n}=s(M^{n})}$

${\displaystyle s(M)=0,M<=1/2,}$${\displaystyle s(M)=M-1/21/2<\vert {}M\vert {}<3/2}$${\displaystyle s(M)=1M>=3/2}$

Here${\displaystyle M_{j+{\frac {1}{2}}}={\frac {2\left\vert {}u_{j+{\frac {1}{2}}}\right\vert {}}{c_{j}+c_{j+1}}}}$

with c as the speed of the sound.

And it is found that here m and p are the terms evaluated at new time level t^(n+1) This is mostly based on the 1 dimension case

Pressure correction method

For a higher order nonlinear system we have to use iterative methods. So for the better results we use the pressure-correction method ^[2] In this method first t^(n+1)is obtained. Next a momentum prediction m* by replacing p^(n+1/2) by p^n

${\displaystyle m_{j+1/2}^{*}-m_{j+{\frac {1}{2}}}^{n}+\lambda {}\left(u^{n}m^{n}+p^{n}\right){\vert {}}_{j}^{j+1}=0}$

A momentum correction${\displaystyle \delta {}m=m^{n+1}-m^{*}}$is postulated as

${\displaystyle \delta {}m_{j+{\frac {1}{2}}}=-\left({\frac {1}{2}}\right)\lambda {}\delta {}p{\vert {}}_{j}^{j+1}}$

${\displaystyle \delta {}p=p^{n+1}-p^{n}}$ Substitution of ${\displaystyle m^{n+1}=m^{*}+\delta {}m}$ gives the following pressure Correction Equation for ${\displaystyle \delta {}p:}$

${\displaystyle \gamma {}M_{r}^{2}\delta {}p_{j}-\left({\frac {1}{2}}\right){\lambda {}}^{2}T_{j}^{n}\left\{\left(1-s_{j+{\frac {1}{2}}}^{n}\right)\delta {}p{\vert {}}_{j}^{j+1}-\left(1-s_{j-{\frac {1}{2}}}^{n}\right)\delta {}p{\vert {}}_{j-1}^{j}\right\}={\rho {}}_{j}^{n}T_{j}{\vert {}}_{n}^{n+1}-\lambda {}T_{j}^{n}\left(s^{n}{\rho {}}^{n}u^{n}\right){\vert {}}_{j-{\frac {1}{2}}}^{j+{\frac {1}{2}}}}$

Boundary conditions

Boundary conditions needed for solving above methods for j=1

${\displaystyle \left({\frac {1}{2}}\right)\lambda {}\delta {}p{\vert {}}_{0}^{1}=-\lambda {}\delta {}m_{1/2}=-\lambda {}({\rho {}}_{b}u_{b}){\vert {}}_{t_{n}}^{t_{n+1}}}$ For j=J the momentum equation is integrated over a half cell:

${\displaystyle m_{J+1/2}^{*}-m_{J+1/2}^{n}+2\lambda {}\left(u^{n}m^{n}+p^{n}\right){\vert {}}_{J}^{J+1/2}=0}$

${\displaystyle p_{J+1/2}^{n}=p_{b}(t^{n})}$${\displaystyle \delta {}m_{J+1/2}=-\lambda {}(p_{b}{\vert {}}_{t_{n}}^{t_{n+1}}-\delta {}p_{j})}$

Runge–Kutta method

There are other methods too for finding the more accurate, more efficient results like one is Runge–Kutta method. ^[3] it is known as a time stepping method in which one can freeze the time of first three steps and jump to the fourth level of the Euler equation with full time T so (m+1) stage becomes: ${\displaystyle T_{j}^{(m+1)}-T_{j}^{n}+{\alpha {}}_{m+1}\lambda {}\left(u^{n}T^{(m)}\right){\vert {}}_{j-1/2}^{j+1/2}+{\alpha {}}_{m+1}\lambda {}\left(\gamma {}-2\right)T_{j}^{(m)}u^{n}{\vert {}}_{j-1/2}^{j+1/2}=0}$

${\displaystyle m_{j+{\frac {1}{2}}}^{\left(m+1\right)}-m_{j+{\frac {1}{2}}}^{n}+{\alpha {}}_{m+1}\lambda {}\left(u^{n}m^{\left(m\right)}+p^{n}\right){\vert {}}_{j}^{j+1}=0}$

In the fourth stage pressure correction is carried out:

${\displaystyle \gamma {}M_{r}^{2}\delta {}p_{j}-\left({\frac {1}{2}}\right){\lambda {}}^{2}T_{j}^{\left(4\right)}\left\{\left(1-s_{j+{\frac {1}{2}}}^{\left(4\right)}\right)\delta {}p{\vert {}}_{j}^{j+1}-\left(1-s_{j-{\frac {1}{2}}}^{\left(4\right)}\right)\delta {}p{\vert {}}_{j-1}^{j}\right\}={\rho {}}_{j}^{n}\left(T_{j}^{\left(4\right)}-T_{j}^{n}\right)-\lambda {}T_{j}^{\left(4\right)}\left(s^{\left(4\right)}{\rho {}}^{n}u^{n}\right){\vert {}}_{j-{\frac {1}{2}}}^{j+{\frac {1}{2}}}}$

References

1. • Eymard, R. Gallouët, T. R., Herbin, R. (2000) The finite volume method Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.
   • LeVeque, Randall (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.
   • Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
2. • M. Thomadakis, M. Leschziner: A PRESSURE-CORRECTION METHOD FOR THE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOWS ON UNSTRUCTURED GRIDS, Int. Journal for Numerical Meth. in Fluids, Vol. 22, 1996
   • A. Meister, J. Struckmeier: Hyperbolic Partial Differential Equations, 1st Edition, Vieweg, 2002
3. • Ascher, Uri M.; Petzold, Linda R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: Society for Industrial and Applied Mathematics, ISBN   978-0-89871-412-8 .
   • Atkinson, Kendall A. (1989), An Introduction to Numerical Analysis (2nd ed.), New York: John Wiley & Sons, ISBN   978-0-471-50023-0 .
   • Butcher, John C. (May 1963), "Coefficients for the study of Runge–Kutta integration processes", Journal of the Australian Mathematical Society, 3 (2): 185–201, doi: 10.1017/S1446788700027932 .