Gas models 37
2.6.3 Inflow/Outflow boundary conditions
The number of variables need to be imposed at the inflow or outflow boundary for a well-posed problem varies depending on whether the flow is locally subsonic or supersonic and is deter- mined from the characteristic theory [74]. In this work mainly supersonic inflow and outflow conditions are considered.
1. Supersonic inflow boundary
According to characteristic theory, for supersonic inflow, all eigenvalues values have the same sign. Therefore at the supersonic inflow boundary, all the flow variables are set to the freestream values corresponding to typical hypersonic flight conditions or ground test conditions.
2. Supersonic outflow boundary
In this case most of the flowfield is assumed to be supersonic at the outflow boundary where it leaves the computational domain. Again this situation also gives all eigenvalues of same sign. Therefore with the consideration of zero upstream influence of supersonic flow, the flow variables at the supersonic outflow boundary are extrapolated from the inte- rior cell. In the present study, a zeroth order extrapolation is used.
Gas models 38
2.7.1 Perfect gas model
This gas model assumes thermally and calorically perfect nature of the flowfield, therfore the ideal gas equation of state can be used to correlate the thermodynamic properties as,
p=ρRT (2.43)
where R is the characteristic gas constant. Further the specific energy e, enthalpy h, and the speed of soundacan be correlated as,
e= RT
(γ−1), h= γRT
(γ−1), a= rγp
ρ (2.44)
whereγis the specific heat ratio for the gas of interest, considered to be 7/5 = 1.4 for air.
After correlating the thermodynamic properties, now let us look in to the calculation of transport properties (k and µ) under this gas model. Since the equations are solved in non- dimensional format, it is not necessary to explicitly calculate the thermal conductivity of the fluid. Instead of that, the Prandtl number needs to be specified in the present solver frame work.
For the perfect gas model the Prandtl number of the fluid is considered as constant. for air as working fluid it is used as 0.71.
Viscosity model
The present solver considers Sutherland’s viscosity model [75] for calculation of coefficient of dynamic viscosity under perfect gas model. The fluid (air) is assumed to be an ideal gas.
Therefore the Sutherland’s model correlates the dynamic viscosity of air with temperature as,
µ=µref
T Tref
3/2
Tref +S T +S
(2.45) where,Tref is the reference temperature, which is taken as 273.15 K for the present study, while µref is the viscosity of air at reference temperature (17.16×10−6N s/m2). In equation (2.45), the Sutherland’s constantS for air is taken as, 110.56.
Gas models 39
2.7.2 Equilibrium flow model
At high speed flows, particularly at hypersonic high enthalpy flows, the perfect gas assumption may not be valid. Presence of dissociation or ionization reactions make this violation. Therefore the chemical reactions involved in the flow field must be modeled to get accurate results. Such high speed chemically reacting flows can be mainly classified into three.
1. Frozen flow: This is one of the extream conditions for the chemically reacting flows. Here the underlying assumption is that, the chemical reactions are exteamily slow so that the high speed flow passing the domain does not feel the effect of chemical reactions. LetTf be the time required for a fluid particle to pass the specified domain at a specified velocity and letTc be the time required for the chemical reaction. Then fluid flow will be in a state of frozen flow, ifTf ≫Tc.
2. Equilibrium flow: In the flowfield, if reactions rate are extreamly high, then chemical reactions take place instantaneously and the flow passing the domain will be in a state of chemical equilibrium. Thus in case of a equilibrium flow Tf ≪ Tc. For a chemical equilibrium flow, the specific heats are the function of both pressure and temperature. The gas constant also vary because of the changes in molecular weight of the mixture.
3. Non-equilibrium flow: The above mentioned flow situations are two extream conditons of hypersonic flow. In reality both of them may not exist. That means the flow will be in a state of reaction. This situation is termed as non-equilibrium flow. For non-equilibrium flow, equation of state is valid, but the gas constant is a variable due to continuously changing molecular weight of the mixture. In order to analyse non-equilibrium reacting flows, one must take care of all the reactions involved in the flow and they are to be modeled accurately.
As an outcome of above discussions it should be noted that non-equilibrium reacting flow models are to be considered in real hypersonic flows. It has been observed that equilibrium flow model can also give accurate results in many hypersonic applications. The general approach to include equilibrium chemistry model needs to solve the governing equtions for the partial pressures of the species. If we restrict our studies to flow involving only air, then we can make
Gas models 40 use of already developed various sets of tables and graphs for equilibrium flow modeling. These graphs and tables are based on the governing equtions for the partial pressures of air. One of such well known tabular approch (Tannehil Muge curve fit) [JC and PH] is utilized in this present solver to model equilibrium flows. The details of that implementation are discussed below.
2.7.2.1 Implementation of Tannehil Mugge curve fit
The thermodynamic properties of equilibrium air can be found out from two known thermo- dynamic variables. Here in this technique density and internal energy are taken as two input thermodynamic variables. Then the solution procedure is followed as below.
1. An effectiveγdenoted asγ¯is first calculated from following equation
¯
γ =a1+a2Y1+a3Z1+a4Y1Z1+a5Y12+a6Z12+a7Y1Z12+a8Z13 + a9+a10Y1+a11Z1+a12Y1Z1
1 + exp [(a13+a14Y1)(Z1+a15Y1+a16)] (2.46) whereY1 = log(ρ/1.292)andZ1 = log(e/78408.4),ρis the density inkg/m3 andeis internal energy inm2/s2
2. Once¯γis calculated, then the equation of state is used to find the pressure.
p=ρe(¯γ−1), (2.47)
wherepis the pressure inN/m2
3. Next the temperature is computed from the following relation
log T
151.78
=b1+b2Y2+b3Z2+b4Y2Z2+b5Y22+b6Z22+b7Y2Z22+b8Z23 +b9+a10Y2+b11Z2+b12Y2Z2+ +b13Z22
1 + exp [(b14Y2+b15)(Z2 +b16)] (2.48) whereY2 = log(ρ/1.225),X2 = log(p/1.0314×105),Z2 =X2−Y2
4. Finally the exact expression for sound speed is calculated from
Temporal discretization 41
a=
"
e (
k1+ ( ¯γ −1)
"
¯ γ+k2
∂γ¯
∂logee
ρ
# +k3
∂γ¯
∂logeρ
e
)#1/2
(2.49)
The coefficients used in above equations are taken from reference [JC and PH].