# Physical Definition

The physical definition of a case includes the definition of the free-stream, the reference values and the non-dimensionalization. SU2 offers different ways of setting and computing this definition. This document gives a short overview on the config options and their physical relation.

## Reference Values

Solver Version
EULER, NAVIER_STOKES, RANS, INC_EULER, INC_NAVIER_STOKES, INC_RANS, FEM_EULER, FEM_NAVIER_STOKES 7.0.0

The following table depicts the reference values used by most of the solvers in SU2. The highlighted variables vary depending on the actual solver and the user input.

Variable Unit Reference
Length $$m$$ $$l_{ref} = 1$$
Density $$\frac{kg}{m^3}$$ $$\rho_{ref}$$ (based on user input)
Velocity $$\frac{m}{s}$$ $$v_{ref}$$ (based on user input)
Temperature $$K$$ $$T_{ref}$$ (based on user input)
Pressure $$Pa$$ $$p_{ref}$$ (based on user input)
Viscosity $$\frac{kg}{ms}$$ $$\mu_{ref} = \rho_{ref}v_{ref}l_{ref}$$
Time $$s$$ $$t_{ref} = \frac{l_{ref}}{v_{ref}}$$
Heatflux $$\frac{W}{m^2}$$ $$Q_{ref} = \rho_{ref}v^3_{ref}$$
Gas Constant $$\frac{m^2}{s^2 K}$$ $$R_{ref} = \frac{v^2_{ref}}{T_{ref}}$$
Conductivity $$\frac{W}{mK}$$ $$k_{ref} = \mu_{ref}R_{ref}$$
Force $$N$$ $$F_{ref} = \rho_{ref}v^2_{ref}l^2_{ref}$$

## Free-Stream Definition (Compressible)

Solver Version
EULER, NAVIER_STOKES, RANS,FEM_EULER, FEM_NAVIER_STOKES 7.0.0

The physical definition for the compressible solvers in SU2 based around the definition of the free-stream. The free-stream values are not only used as boundary conditions for the MARKER_FAR option, but also for initialization and non-dimensionalization. That means even if you don’t have any farfield BCs in your problem, it might be important to prescribe physically meaningful values for the options.

### Thermodynamic State

The thermodynamic state of the free-stream is defined by the pressure $$p_{\infty}$$, the density $$\rho_{\infty}$$ and the temperature $$T_{\infty}$$. Since these quantities are not independent, only two of these values have to be described and the third one can be computed by an equation of state, depending on the fluid model used. There are two possible ways implemented that can be set using FREESTREAM_OPTION:

• TEMPERATURE_FS (default): Density $$\rho_{\infty}$$ is computed using the specified pressure $$p_{\infty}$$ (FREESTREAM_PRESSURE) and temperature $$T_{\infty}$$ (FREESTREAM_TEMPERATURE).
• DENSITY_FS: Temperature $$T_{\infty}$$ is computed using the specified pressure $$p_{\infty}$$ (FREESTREAM_PRESSURE) and density $$\rho_{\infty}$$ (FREESTREAM_DENSITY).

### Mach Number and Velocity

The free-stream velocity $$v_{\infty}$$ is always computed from the specified Mach number $$Ma_{\infty}$$ (MACH_NUMBER) and the computed thermodynamic state. The flow direction is based on the angle of attack (AOA) and the side-slip angle (SIDESLIP_ANGLE, for 3D).

### Reynolds Number and Viscosity

If it is a viscous computation, by default the pressure $$p_{\infty}$$ will be recomputed from a density $$\rho_{\infty}$$ that is found from the specified Reynolds number $$Re$$ (REYNOLDS_NUMBER). Note that for an ideal gas this does not change the Mach number $$Ma_{\infty}$$ as it is only a function of the temperature $$T_{\infty}$$. If you still want to use the thermodynamic state for the free-stream definition, set the option INIT_OPTION to TD_CONDITIONS (default: REYNOLDS). In both cases, the viscosity is computed from the dimensional version of Sutherland’s law or the constant viscosity (FREESTREAM_VISCOSITY), depending on the VISCOSITY_MODEL option.

### Non-Dimensionalization

For all schemes, as reference values for the density and temperature the free-stream values are used, i.e. $$\rho_{ref} = \rho_{\infty}, T_{ref} = T_{\infty}$$. The reference velocity is based on the speed of sound defined by the reference state: $$v_{ref} = \sqrt{\frac{p_{ref}}{\rho_{ref}}}$$. The dimensionalization scheme can be set using the option REF_DIMENSIONALIZATION and defines how the reference pressure $$p_{ref}$$ is computed:

• DIMENSIONAL: All reference values are set to 1.0, i.e. the computation is dimensional.
• FREESTREAM_PRESS_EQ_ONE: Reference pressure equals free-stream pressure, $$p_{ref} = p_{\infty}$$.
• FREESTREAM_VEL_EQ_MACH: Reference pressure is chosen such that the non-dimensional free-stream velocity equals the Mach number: $$p_{ref} = \gamma p_{\infty}$$.
• FREESTREAM_VEL_EQ_ONE: Reference pressure is chosen such that the non-dimensional free-stream velocity equals 1.0: $$p_{ref} = Ma^2_{\infty} \gamma p_{\infty}$$.

## Flow Condition (Incompressible)

Solver Version
INC_EULER, INC_NAVIER_STOKES, INC_RANS 7.0.0

The physical definition of the incompressible solvers is accomplished by setting an appropriate flow condition for initialization and non-dimensionalization. SU2 solves the incompressible Navier-Stokes equations in a general form allowing for variable density due to heat transfer through the low-Mach approximation (or incompressible ideal gas formulation).

### Thermodynamic and Gauge Pressure

In the incompressible problem the thermodynamic pressure is decoupled from the governing equations and density is therefore only a function of temperature variations. The absolute value of the pressure is not important and any reference to the pressure $$p$$ is considered as the gauge value, i.e. it is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus (an arbitrary) atmospheric pressure.

### Initial State and Non-Dimensionalization

The initial state, i.e. the initial values of density $$\rho_0$$, velocity vector $$\bar{v}_{0}$$ and temperature $$T_0$$ are set with INC_DENSITY_INIT, INC_VELOCITY_INIT and INC_TEMPERATURE_INIT, respectively. The initial pressure $$p_0$$ is always set to 0.0.

The reference values $$\rho_{ref}, T_{ref}, v_{ref}$$ equal the initial state values by default (or if INC_NONDIM= INITIAL_VALUES). If INC_NONDIM is set to REFERENCE_VALUES you can define different values for them using the options INC_DENSITY_REF, INC_VELOCITY_REF and INC_TEMPERATURE_REF. The reference pressure is always computed by $$p_{ref} = \rho_{ref}v^2_{ref}$$.

Note: The initial state is also used as boundary conditions for MARKER_FAR.