2D Unstructured Euler Solver

Department of Aeronautics and Astronautics

Stanford University

AA-215B: Advanced CFDProfessor A. Jameson

Britton J. Olson

May 24, 2010

Abstract

In Aeronautical Engineering, the ability to solve the governing equationsof motion in a complex domain is vital. In a general sense, unstructuredsolvers can provide robust treatment of nearly any boundary which mightrepresent some engineering interest. For design purposes, the wave drag ordrag produces by shock waves is of interest in designing airfoil shapes whichare optimal. Neglecting viscous effect, the full Navier-Stokes equations re-duce to the Euler equations which capture all the compressibility effects oftransonic flow. (eg Shock waves, Prandlt-Meyer expansion fans, entropywaves, etc.) In this brief note, we report our method and implementationof a simple low-order unstructured solver of the Euler equations. We testour code on 2 simple test cases whose results are given in a latter section.The solutions converge to steady state in a reasonable amount of time andgiven insight on the physics of the compressible equations of motion. Sucha solver, might be used to produce or validate aerodynamic design.

1 Algorithm

A node based edge flux method was used to solve the 2D Euler equations. In thisscheme, the conserved variables (ρ, ρu, ρv, and ρE) are stored at each node whilefluxes are computed across edges. The data is stored so that each edge containsa list of 4 nodes listed in a consistent manner (counter-clockwise in this case). Ateach iteration, a loop over all edges was performed to find the total flux residualat each node. Using the example in figure 1, the flux across edge 24 is computedby averaging the flow variables at nodes 2 and 4. The flux contribution is thenapplied to nodes 1 and 3 across from the edge.

Figure 1: Node based edge flux method.

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At free stream boundaries, node values are set to the prescribed free streamvalues. At the wall boundary, the flux computation is done similarly as with inte-rior edges, except flow tangency is enforced. Additionally, to maintain consitencyin the control volume, the flux on a wall boundary edge is applied to the wallnodes as well. For interior nodes, the boundaries of the control volume surroundthe node (see figure 2) so the scheme described above will ensure proper conserva-tion at each node. However, at wall edges, a node resides on the control volumeboundary (see figure 3). As a result, the computed wall flux must be applied tothe wall node as well to maintain conservation.

Figure 2: Control volume around an interior node.

Figure 3: Control volume around a boundary node.

In order to maintain stability in flows with shocks, a scalar artificial diffusionwas added. The scalar diffusion was taken as the maximum eigenvalue of the Roematrix and is applied everywhere in the flow. Although this method works, it isquite diffuse and leads to smeared shocks. There are two major issues with this

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method in terms of accuracy. First, the highest value of the Roe diffusion is usedeverywhere since the maximum eigenvalue of the Roe matrix is taken as the scalar.This leads to unncesssary diffusion in the majority of the domain. In addition, theartificial field is applied everywhere, whereas a more sophisticated method woulddetect and apply diffusion only to necessary areas (namely at shocks).

2 Governing Equations

We provide here, for reference, the governing equations. 2-D Euler equations(conservation of mass, momenta and energy) can be simply written as:

∂U

∂t+∇ · Fx = 0 (1)

U =

ρρuρvρE

,F x =

ρu

u2 + puv

u(ρE + p)

,F y =

ρvuv

v2 + pv(ρE + p)

, (2)

E =p

ρ(γ − 1)+

1

2uiui (3)

We solve then using a simple integration and in weak form as:

∂

∂t

∫Udt+

∮(F xdy − F ydx) = 0 (4)

We can discretely solve this by summing the edge contributions to each nodek as:

(∑k

A

)∂Un

∂t+

1

2

∑k

[(F k+1

x + F kx)(yk+1 − yk)− (F k+1

y + F ky)(xk+1 − xk)

]= 0

(5)Were k are the nodes which surround a particular point n, as in figure 2 and S

is the area. We the above approximations, we can now readily solve the discreteapproximation to the Euler equation. The source code (written in fortran 90) canbe provided by means a simple svn repository checkout by issuing: “svn checkouthttp://cybo.googlecode.com/svn/trunk/ euler2d”.

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3 Results

3.1 Circular Arc

Transonic flow (M∞ = 0.9) over a 5% thick (at 50% chord) circular arc bump wascomputed. Results showing contours of Mach number and pressure are below. Inboth plots, a supersonic region is visible over the bump (although the shock isvisibly diffuse due to the use of first order scalar artificial diffusion). The surfacepressure distribution shows that the shock is captured in approximately 5 points.Since the method is low order (especially the shock capturing scheme), a fairly finetriangulation was used.

(a) (b)

−0.5 0 0.5 1 1.5

−0.5

0

0.5

1

X−axis

−C

p

(c) (d)

Figure 4: 5% Cicular arc foil. M∞ = 0.9

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3.2 NACA 0012

Transonic flow (M∞ = 0.75) over a NACA 0012 airfoil at α = 10◦ was alsocomputed. Contours of Mach number and pressure show a supersonic region onthe upper surface of the airfoil. Again, due to the lower order of the method arelatively fine mesh was used to allow for a reasonable shock.

(a) (b)

−0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

1.5

X−axis

−C

p

(c) (d)

Figure 5: NACA 0012. M∞ = 0.75, α = 10◦.

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