View source for Unified methods for computing incompressible and compressible flow

{{Proposed deletion/dated
|concern = This article seems like a mix of original research and/or an instruction manual, which are not allowed on Wikipedia. Also, as it stands, the article would need complete rewriting, so I propose deleting per WP:TNT
|timestamp = 20250301205824
|nom = 7804j
|help = off
}}
{{Prod2|Agreed: it both [[WP: SYNTH|is synthesis]] and [[WP:TNT|must be completely re-written to be an article]]. It's great to show all these formulae in one place, but there are categories for that.}}
{{Technical|date=June 2023}}
Computation of [[incompressible flow|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 [[Navier–Stokes equations#Incompressible flow|incompressible Navier Stokes]] and [[Euler equation]]s 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 ==
<math>\frac{\partial{}\rho{}}{\partial{}t}+\nabla{}.\left(\rho{}V\right)=0
</math>
== Conservation of momentum ==
<math>\rho{}\frac{DV}{Dt}=-\nabla{}p+\nabla{}.\tau{}+\rho{}f
</math>
== Conservation of energy ==
<math>\rho{}\left[\frac{\partial{}h}{\partial{}t}+\nabla{}.\left(hV\right)\right]=-\frac{Dp}{Dt}+\nabla{}.\left(k\nabla{}T\right)+\Phi{}</math>
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 scheme|MUSCL]] and [[Runge–Kutta]] time stepping
== Governing equation ==
=== Conservation of mass ===
<math>{\rho{}}_pp_t+{\rho{}}_TT_t+m_x=0
</math>
=== Equation of state ===
<math>{\rho{}}_p=\gamma{}M_r^2/T
</math>
<math>{\rho{}}_T=-\frac{\rho{}}{T}</math>
=== Momentum equation ===
<math>m_t+{(um+p)}_x=0
</math>
By using the dimensionless pressure and equation of state the governing equation can be best described as:
<math>T_t+{(uT)}_x+\left(\gamma{}-2\right)Tu_x=0
</math>
== Finite volume scheme ==
For the above specified governing equations the [[Finite volume method|finite volume scheme]]<ref>*Eymard, R. Gallouët, T. R., [[Raphaèle Herbin|Herbin, R.]] (2000) ''The finite volume method''  Handbook of Numerical Analysis, Vol. VII, 2000, p.&nbsp;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.</ref> is
<math>\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{}}^nu^n+\left(1-s^n\right)m^{n+1}\right){\vert{}}_{j-1/2}^{j+1/2}=0</math>
<math>m_{j+1/2}^{n+1}-m_{j+1/2}^n+\lambda{}\left(u^nm^n+p^{n+1/2}\right){\vert{}}_j^{j+1}=0
</math>
<math>T_j^{n+1}-T_j^n+\lambda{}\left(u^nT^n\right){\vert{}}_{j-\frac{1}{2}}^{j+\frac{1}{2}}+\lambda{}\left(\gamma{}-2\right)T_j^nu^n{\vert{}}_{j-\frac{1}{2}}^{j+\frac{1}{2}}=0
</math>
where
<math>\lambda{}=\tau{}/h
</math>
<math>p^{n+1/2}=(p^n+p^{n+1})/2</math>
<math>s^n=s(M^n)</math>
<math>s(M)=0       ,M<=1/2,
</math>
<math>s(M)=M-1/2           1/2<\vert{}M\vert{} <3/2</math>
<math>s(M)=1                M>=3/2
</math>
Here<math>M_{j+\frac{1}{2}}=\frac{2\left\vert{}u_{j+\frac{1}{2}}\right\vert{}}{c_j+c_{j+1}}</math>
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]]<ref>*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</ref>
In this method first t^(n+1)is obtained. Next a momentum prediction m* by replacing p^(n+1/2) by p^n
<math>m_{j+1/2}^*-m_{j+\frac{1}{2}}^n+\lambda{}\left(u^nm^n+p^n\right){\vert{}}_j^{j+1}=0
</math>
A momentum correction<math> \delta{}m=m^{n+1}-m^*</math>is postulated as
<math>\delta{}m_{j+\frac{1}{2}}=-\left(\frac{1}{2}\right)\lambda{}\delta{}p{\vert{}}_j^{j+1}
</math>
<math>\delta{}p=p^{n+1}-p^n
</math>
Substitution of <math>m^{n+1}=m^*+\delta{}m</math> gives the following pressure
Correction Equation for <math>\delta{}p:</math>
<math>\gamma{}M_r^2\delta{}p_j-\left(\frac{1}{2}\right){\lambda{}}^2T_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^nT_j{\vert{}}_n^{n+1}-\lambda{}T_j^n\left(s^n{\rho{}}^nu^n\right){\vert{}}_{j-\frac{1}{2}}^{j+\frac{1}{2}}
</math>
=== Boundary conditions ===
Boundary conditions needed for solving above methods for j=1
<math>\left(\frac{1}{2}\right)\lambda{}\delta{}p{\vert{}}_0^1=-\lambda{}\delta{}m_{1/2}=-\lambda{}({\rho{}}_bu_b){\vert{}}_{t_n}^{t_{n+1}}
</math>
For j=J the momentum equation is integrated over a half cell:
<math>m_{J+1/2}^*-m_{J+1/2}^n+2\lambda{}\left(u^nm^n+p^n\right){\vert{}}_J^{J+1/2}=0
</math>
<math>p_{J+1/2}^n=p_b(t^n)
</math>
<math>\delta{}m_{J+1/2}=-\lambda{}(p_b{\vert{}}_{t_n}^{t_{n+1}}-\delta{}p_j)
</math>
== Runge–Kutta method ==
There are other methods too for finding the more accurate, more efficient results like one is Runge–Kutta method.<ref>*{{Citation | last1=Ascher | first1=Uri M. | last2=Petzold | first2=Linda R. |author2-link=Linda Petzold| title=Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | isbn=978-0-89871-412-8 | year=1998}}.
* {{Citation | last1=Atkinson | first1=Kendall A. | title=An Introduction to Numerical Analysis | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-50023-0 | year=1989}}.
* {{Citation| last1 = Butcher | first1 = John C. | author1-link=John C. Butcher | title=Coefficients for the study of Runge–Kutta integration processes | journal = Journal of the Australian Mathematical Society | doi=10.1017/S1446788700027932 | volume=3 | issue=2 |date=May 1963 | pages=185–201| doi-access=free }}.</ref>  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:
<math>T_j^{(m+1)}-T_j^n+{\alpha{}}_{m+1}\lambda{}\left(u^nT^{(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
</math>
<math>m_{j+\frac{1}{2}}^{\left(m+1\right)}-m_{j+\frac{1}{2}}^n+{\alpha{}}_{m+1}\lambda{}\left(u^nm^{\left(m\right)}+p^n\right){\vert{}}_j^{j+1}=0
</math>
In the fourth stage pressure correction is carried out:
<math>\gamma{}M_r^2\delta{}p_j-\left(\frac{1}{2}\right){\lambda{}}^2T_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{}}^nu^n\right){\vert{}}_{j-\frac{1}{2}}^{j+\frac{1}{2}}</math>
==References==
{{Reflist}}
[[Category:Computational fluid dynamics]]