Next: Laminar viscous compressible compression Up: Simple example problems Previous: Transient laminar incompressible Couette Contents

## Stationary laminar inviscid compressible airfoil flow

Figure 34: Mesh for the naca012 airfoil flow

Figure 35: Mach number in the naca012 airfoil flow

Figure 36: Pressure coefficient in the naca012 airfoil flow

In [55] the results of CFD-calculations for several airfoils are reported. Here, the computations for (Mach number at infinity) and (angle of attack) are reported. The input deck for this calculation can be found in the fluid examples test suite (agard05.inp).

To explain the differences in the input deck between incompressible and compressible flow the crucial section from the compressible input deck is reproduced below.

    *EQUATION
    2
    3,2,-0.99030509E+00,3,1,-0.13890940E+00
    2
    3756,2,-0.99030509E+00,3756,1,-0.13890940E+00
    ...
    *MATERIAL,NAME=AIR
    *CONDUCTIVITY
    0.
    *FLUID CONSTANTS
    1.,1.d-20,293.
    *SPECIFIC GAS CONSTANT
    0.285714286d0
    *SOLID SECTION,ELSET=Eall,MATERIAL=AIR
    *PHYSICAL CONSTANTS,ABSOLUTE ZERO=0.
    *INITIAL CONDITIONS,TYPE=FLUID VELOCITY
    Nall,1,0.99254615
    Nall,2,0.12186934
    Nall,3,0.d0
    *INITIAL CONDITIONS,TYPE=PRESSURE
    Nall,0.49603175
    *INITIAL CONDITIONS,TYPE=TEMPERATURE
    Nall,1.73611111
    *VALUES AT INFINITY
    1.73611111,1.,0.49603175,1.,1.
    **
    *STEP,INCF=40000,SHOCK SMOOTHING=0.1
    *STATIC,EXPLICIT
    1.,1.
    *BOUNDARY
    BOU1,11,11,1.73611111
    BOU1,1,1,0.99254615
    BOU1,2,2,0.12186934
    BOU1,8,8,0.49603175
    Nall,3,3,0.
    *NODE FILE,FREQUENCYF=40000
    V,PS,CP,TS,TT,MACH
    *END STEP

Since for compressible flow the temperature, velocity and pressure are linked through the ideal gas equation, the definition of the thermal conductivity and specific heat is mandatory. Inviscid flow was triggered by the definition of a very low viscosity AND slip boundary conditions at the airfoil surface through equations. The specific gas constant is defined with the appopriate keyword. It only depends on the kind of gas and not on the temperature. The physical constants card is used to define absolute zero for the temperature scale. This information is needed since the temperature in the gas equation must be specified in Kelvin. Initial conditions must be specified for the velocity, pressure and temperature. Careful selection of these values can shorten the computational time. The values at infinity (defined with the *VALUES AT INFINITY card) are used to calculate the pressure coefficient. In viscous calculations they are used for the computation of the friction coefficient too. The smoothing parameter on the *STEP card is used to define shock smoothing and will be discussed in the next paragraph. Finally, compressible calculations are performed explicitly. Therefore, the EXPLICIT parameter on the *STATIC or *DYNAMIC keyword is mandatory. It is the EXPLICIT parameter which tells CalculiX whether the flow is compressible or incompressible. With the EXPLICIT parameter the flow is assumed to be compressible, else it is assumed to be incompressible. The use of the *STATIC keyword tells CalculiX that the calculation is stationary. Instationary calculations are triggered with the *DYNAMIC keyword. In reality, all CFD-calculations in CalculiX are instationary. The *STATIC keyword, however, forces the calculations to be pursued until steady state is reached (so the time used is virtual). Dynamic calculations stop as soon as the final time is reached (the time is real).

In compressible calculations shock smoothing is frequently needed in order to avoid divergence. Shock smoothing, however, can change the solution. Therefore, the shock smoothing coefficient, which can take values between 0. and 2., should be chosen as small as possible. For the agard05 example a value of 0.1 was needed. In general, additional viscosity will reduced the shock smoothing needed to avoid divergence. There is a second effect of the shock smoothing coefficient: there is no clear steady state convergence any more. In order to understand this some additional information about the way CFD-calculations in CalculiX are performed. The initial increment size which is specified by the user underneath the *STATIC or *DYNAMIC card is a mechanical increment size. For each mechanical increment an instationary CFD-calculation is performed subject to the actual loads (up to steady state for a *STATIC calculation). For this CFD-calculations subincrements are used, the size of which depends on the physical characteristics of the flow (viscosity, heat conductivity etc.). They are determined such that stability is assured (or at least very likely). In CalculiX, steady state convergence is detected as soon as the change in the conservative variables ( etc.) from subincrement to subincrement does not exceed times the actual values of these variables. In calculations with a nonzero shock smoothing coefficient the change in variables at first decreases down to a certain level about which it oscillates erraticaly. In that case, steady state is detected as soon as the tangent of a linear regression curve through the last half of the change in variables values drops below a given number. The change in the conservative variables is stored in a file with the name jobname.cvg. The user may force convergence by limiting the number of subincrements with the INCF parameter on the *STEP card. As soon as INCF subincrements are calculated the CFD-calculation is assumed to be finished and the next mechanical increment is started.

Figure 34 shows the mesh used for the agard05 calculation. It consists of linear wedge elements. In CalculiX, only linear elements (tetrahedra, hexahedra or wedges) are allowed for CFD-calculations. It is finer along the airfoil (but not as fine as needed to capture the boundary layer in viscous calculations). Figures 35 and 36 shows the Mach number and the pressure coefficient, respectively. The maximum Mach number in [55] is about 1.78, the maximum pressure coefficient is about -0.55. This agrees well with the present results. Increasing the shock smoothing coefficient leads to smoothing fringe plots, however, the actual values become worse.

Next: Laminar viscous compressible compression Up: Simple example problems Previous: Transient laminar incompressible Couette Contents