7.14.2.1. Aerodynamic Convection Heat Transfer

High-speed flows about bodies are often characterized by large temperature gradients and large variations of the flow properties through the boundary layer. The problem of heat transfer about the body is often addressed using numerical techniques of CFD to resolve the thermal transport through the boundary layer. Because CFD resources may not be readily applied at the design stage researchers often apply simplifying assumptions to arrive at alternative models for solving the problem of heat transfer about bodies exposed to high-speed flow. One common modeling approach is to define a convection coefficient and reference temperature that depend upon flight conditions. The aero heat flux boundary condition provides a means for supplying details of the flight conditions to arrive at a representative convection coefficient and reference temperature for two heat flux models, one derived by [58] and one that varies only with altitude.

The model by Eckert suggests that a convective heat transfer approach ignoring boundary layer temperature gradients may be used if the properties are evaluated at an alternative reference temperature representative of the averaged flow conditions. Syntax for typical usage of this model is shown below.

Begin AERO HEAT FLUX BOUNDARY CONDITION body_heating
  START TIME =  0.0
  STOP TIME =  20.0
  freestream temperature = 233.15
  freestream pressure    = 5.066e3
  mach number = 3.0
  fluid viscosity temperature function            = air_viscosity
  fluid specific heat temperature function        = air_specheat
  fluid thermal conductivity temperature function = air_cond
  fluid gamma = 1.4
  adiabatic wall temperature from recovery factor
  eckert convective coefficient
  COORDINATE OFFSET AXIS = x
  cone length = 0.0
  add surface surface_1
  integrated power output total_heating
End

Another aerodynamic heat flux model defines a convective coefficient that relies upon the variation of density with altitude. Syntax for typical usage of this model is shown below.

Begin AERO HEAT FLUX BOUNDARY CONDITION body_heating
  START TIME =  0.0
  STOP TIME =  20.0
  Density Ratio Convective Coefficient
  Freestream Density Altitude Function = density_func
  Altitude time function = altitude_time_func
  Reference Density = 1.15
  Reference HTC = 25.0
  Density Ratio exponent = 0.6
  reference temperature time function = ref_temp_func
  add surface surface_1
  integrated power output total_heating
End

A more complete description of syntax for the aero heating models is described below

7.14.2.1.1. Aero Heat Flux Boundary Condition

Scope: Aria Region, Equation System, Explicit Equation System, Root Finder Equation System
Summary: This command block defines a distributed convective heat flux about a body based upon aerodynamic flight conditions applied on the given surface.
Description: The aerodynamic heat flux depends upon characterization of flight conditions prescribed using data tables and spatial location on the body of interest.

begin Aero Heat Flux Boundary Condition Name

   Add Surface SurfaceList...

   Adiabatic Wall Temperature {=} Value

   Adiabatic Wall Temperature Altitude Function {=} Function

   Adiabatic Wall Temperature From Recovery Factor

   Adiabatic Wall Temperature Time Function {=} Function

   Altitude {=} Value

   Altitude Time Function {=} Function

   Cone Length {=} Value

   Coordinate Offset {=} Value

   Coordinate Offset Axis {=} axis

   Density Ratio Convective Coefficient

   Density Ratio Exponent {=} Value

   Eckert Convective Coefficient

   Fluid Gamma {=} Value

   Fluid Gas Constant {=} Value

   Fluid Properties Temperature Function {=} Function

   Fluid Specific Heat Temperature Function {=} Function

   Fluid Thermal Conductivity Temperature Function {=} Function

   Fluid Viscosity Temperature Function {=} Function

   Freestream Density Altitude Function {=} Function

   Freestream Pressure {=} Value

   Freestream Pressure Altitude Function {=} Function

   Freestream Pressure Time Function {=} Function

   Freestream Temperature {=} Value

   Freestream Temperature Altitude Function {=} Function

   Freestream Temperature Time Function {=} Function

   Integrated Flux Output VariableName

   Integrated Power Output VariableName

   Mach Number {=} Value

   Mach Number Time Function {=} Function

   Reference Density {=} Value

   Reference Htc {=} Value

   Reference Temperature {=} Value

   Reference Temperature Time Function {=} FunctionName

   Start Time {=} Value

   Stop Time {=} Value

   Uq Flux Multiplier {=} Value

   Use Toggle Block ToggleName [ {on} ElementBlockList...  ]

end Aero Heat Flux Boundary Condition Name

Line Commands

Add Surface

Syntax: Add Surface SurfaceList…
Summary: Adds surfaces, by name, to a boundary condition’s extent.
Description: This line command is used to add surfaces to the extent of a boundary condition. In Exodus II, surfaces are specified as side sets, that have a global integer identifier. For example, side set 12 would be added by this line command using the surface name surface_12. Note that in SIERRA, each element of an array of strings must be separated by whitespace.

Parameter	Value	Default
SurfaceList	string…	–

Adiabatic Wall Temperature

Syntax

Adiabatic Wall Temperature {=} Value

Summary

Specifies a constant adiabatic wall temperature. Usage: ECKERT CONVECTIVE COEFFICIENT.

FREESTREAM TEMPERATURE and fluid properties consistent with this ADIABATIC WALL TEMPERATURE must also be provided.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Adiabatic Wall Temperature Altitude Function

Syntax

Adiabatic Wall Temperature Altitude Function {=} Function

Summary

Specifies the adiabatic wall temperature versus altitude function for climate condition. Usage: ECKERT CONVECTIVE COEFFICIENT.

FREESTREAM TEMPERATURE and fluid properties consistent with this ADIABATIC WALL TEMPERATURE must also be provided.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Adiabatic Wall Temperature From Recovery Factor

Syntax

Adiabatic Wall Temperature From Recovery Factor

Summary

Specifies that the adiabatic wall temperature be computed using the recovery factor, the stagnation temperature and the freestream temperature. Usage: ECKERT CONVECTIVE COEFFICIENT.

Description

The adiabatic wall temperature $T_{aw}$ is computed as

$T_{aw} = r T_{o} + ( 1 - r ) T_{\infty}$

where $r$ is the recovery factor, $T_{o}$ is the stagnation temperature and $T_{\infty}$ is the freestream temperature. The user must provide specification of MACH NUMBER, FREESTREAM TEMPERATURE, FREESTREAM PRESSURE and FLUID_GAMMA models. Stagnation temperature and flow velocity $v$ will be internally computed from the MACH NUMBER. The recovery factor is given in terms of the freestream Prandtl number $C_{p} \mu / K$ as

$r = \left\{ \begin{array}{l} Pr^{1/2} \;\;\; \mbox{laminar flow} \\ Pr^{1/3} \;\;\; \mbox{turbulent flow} \;\; Re > 5 X 10^5 \end{array} \right. \;.$

Hence one must provide functions for specific heat, dynamic viscosity and thermal conductivity as a function of temperature. The flow regime varies spatially in the COORDINATE OFFSET direction as determined using the local Reynolds number

$Re_{x} = \frac{\rho v x}{\mu} \;.$

Here the pressure $P$ will vary with altitude hence density is evaluated using the ideal gas law

$\rho = \frac{P}{R T_{\infty}}$

Adiabatic Wall Temperature Time Function

Syntax

Adiabatic Wall Temperature Time Function {=} Function

Summary

Specifies the name of adiabatic wall temperature versus time function. Usage: ECKERT CONVECTIVE COEFFICIENT.

FREESTREAM TEMPERATURE and fluid properties consistent with this ADIABATIC WALL TEMPERATURE must also be provided.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Altitude

Syntax

Altitude {=} Value

Summary

Specifies a constant altitude.

Used in conjunction with FREESTREAM TEMPERATURE ALTITUDE FUNCTION, FREESTREAM TEMPERATURE FUNCTION and ECKERT CONVECTIVE COEFFICIENT. Alternative usage with FREESTREAM DENSITY ALTITUDE FUNCTION and DENSITY RATIO CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Altitude Time Function

Syntax

Altitude Time Function {=} Function

Summary

Specifies the altitude versus time function for a given flight. The tabulated time is defined relative to the START TIME.

Used in conjunction with FREESTREAM TEMPERATURE ALTITUDE FUNCTION, FREESTREAM TEMPERATURE FUNCTION and ECKERT CONVECTIVE COEFFICIENT. Alternative usage with FREESTREAM DENSITY ALTITUDE FUNCTION and DENSITY RATIO CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Cone Length

Syntax: Cone Length {=} Value
Summary: Specifies the length at which the body profile transitions to a constant diameter. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Coordinate Offset

Syntax: Coordinate Offset {=} Value
Summary: Specifies the coordinate offset from which the spatial position will be defined in the ECKERT CONVECTIVE COEFFICIENT calculations $x = X - \mbox{offset}$ where X is the Cartesian model coordinate from the mesh prescribed using COORDINATE OFFSET AXIS and is assumed to define the leading edge. The value of $x$ is used in computation of the Reynolds number $Re^{*}_{x} = \frac{\rho^{*} v x}{\mu}$ hence the evaluation of $x$ must provide values greater than zero.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Coordinate Offset Axis

Syntax

Coordinate Offset Axis {=} axis

Summary

Specifies the Cartesian coordinate axis (X, Y or Z) from which spatial position will be defined when using the ECKERT CONVECTIVE COEFFICIENT. The surface heat transfer coefficient will be computed spatially relative to zero on this axis.

For models in which the geometry is offset from a zero reference one must define the COORDINATE OFFSET.

Parameter	Value	Default
{=}	{= \| are \| is}	–
axis	string	none

Density Ratio Convective Coefficient

Syntax

Density Ratio Convective Coefficient

Summary

Specifies a density ratio heat transfer coefficient model. Requires that FREESTREAM DENSITY ALTITUDE FUNCTION, REFERENCE DENSITY, REFERENCE HTC, DENSITY RATIO EXPONENT, START TIME and STOP TIME specifications be supplied.

Description

The heat transfer coefficient that varies as

$h = h_{o} \left[ \frac{\rho(A)}{\rho_{o}} \right]^{n}$

where $h_{o}$ is the reference heat transfer coefficient, $\rho(A)$ is the altitude dependent density, $\rho_{o}$ is the reference density and $n$ is the density ratio exponent. Here the model parameters $h_{o}$ , $\rho_{o}$ and $n$ are calibrated based upon measured temperature data.

Density Ratio Exponent

Syntax: Density Ratio Exponent {=} Value
Summary: Specifies the density ratio exponent value for the density ratio heat transfer coefficient model. Usage: DENSITY RATIO CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Eckert Convective Coefficient

Syntax

Eckert Convective Coefficient

Summary

Specifies use of the Eckert heat transfer coefficient model for high-speed flow. Requires that freestream FLUID PROPERTY TEMPERATURE FUNCTIONs, START TIME, STOP TIME, FREESTREAM PRESSURE and ADIABATIC WALL TEMPERATURE specifications be supplied.

Description

A constant property heat transfer coefficient approach in which properties are evaluated at an alternative temperature

$T^{*} = T_{\infty} + 0.5 ( T - T_{\infty} ) + 0.22 ( T_{aw} - T_{\infty} )$

where $T_{\infty}$ is the freestream temperature, $T$ is the surface temperature, $T_{aw}$ is the adiabatic wall temperature. $T_{aw}$ will vary spatially depending upon the local Reynolds number

$Re^{*}_{x} = \frac{\rho^{*} v x}{\mu}$

where $\mu$ is evaluated at $T^{*}$ and the coordinate $x > 0$ . The FREESTREAM PRESSURE varies with altitude hence the density $\rho^{*}$ is evaluated using the ideal gas law

$\rho^{*} = \frac{P}{R T^{*}} \;.$

The Stanton number represents the ratio of heat transferred into the fluid flow to the thermal capacity of the fluid. By developing the relationship between fluid friction and heat transfer frictional resistance, the heat transfer can be expressed in terms of the Stanton number to arrive at an approximate heat transfer coefficient for various portions of the flow regime

$St^{*}_{x} Pr^{*2/3} = \begin{cases} 0.332 ( Re_{x}^{*}/ f_{M} )^{-1/2} & Re^{*}_{x} \lt 5 \times 10^{5} \text{ laminar} \\ 0.0296 ( Re_{x}^{*} /f_{M} )^{-1/5} & 5 \times 10^{5} \lt Re^{*}_{x} \lt 10^{7} \text{ turbulent} \\ 0.185 [log10(Re^{*}_{x} /f_{M}) ]^{-2.584} & 10^{7} \lt Re^{*}_{x} \lt 10^{9} \text{ turbulent} \\ 0.00063 & Re^{*}_{x} \gt 10^{9} \text{ turbulent} \end{cases}$

The parameter $f_{M}$ is the Mangler transformation for cone geometry

$f_{M} = \begin{cases} 3 & \text{cone geometry, laminar} \\ 2 & \text{cone geometry, turbulent} \\ 1 & \text{flat plate geometry, laminar} \end{cases}$

While Eckert’s approach considers an average heat transfer coefficient over portions of a surface, here we consider a local heat transfer coefficient, $h(x)$ based upon the above relations that can be applied directly to the surface discretization

$h(x) = \rho^{*} C^{*}_{p} U_{\infty} St^{*}_{x}$

Fluid Gamma

Syntax

Fluid Gamma {=} Value

Summary

Specifies a constant specific heat ratio of the fluid medium. Usage: ECKERT CONVECTIVE COEFFICIENT.

Only constant values of $\gamma$ are allowed.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	1.4

Fluid Gas Constant

Syntax: Fluid Gas Constant {=} Value
Summary: Specifies the gas constant of the fluid medium. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	none

Fluid Properties Temperature Function

Syntax: Fluid Properties Temperature Function {=} Function
Summary: Specifies a multi-column function containing fluid properties density, specific heat, conductivity and dynamic viscosity versus temperature with respective property columns named CP, K and MU. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Fluid Specific Heat Temperature Function

Syntax: Fluid Specific Heat Temperature Function {=} Function
Summary: Specifies the name of a user tabular function for specific heat as a function of temperature. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Fluid Thermal Conductivity Temperature Function

Syntax: Fluid Thermal Conductivity Temperature Function {=} Function
Summary: Specifies the name of a user tabular function for thermal conductivity as a function of temperature. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Fluid Viscosity Temperature Function

Syntax: Fluid Viscosity Temperature Function {=} Function
Summary: Specifies the name of a user tabular function for dynamic viscosity as a function of temperature. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Freestream Density Altitude Function

Syntax

Freestream Density Altitude Function {=} Function

Summary

Specifies the freestream density versus altitude function for climate condition. Usage: DENSITY RATIO CONVECTIVE COEFFICIENT.

Requires that one supply an ALTITUDE model.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Freestream Pressure

Syntax: Freestream Pressure {=} Value
Summary: Specifies a constant freestream pressure.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Freestream Pressure Altitude Function

Syntax: Freestream Pressure Altitude Function {=} Function
Summary: Specifies the freestream pressure versus altitude function for climate condition.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Freestream Pressure Time Function

Syntax: Freestream Pressure Time Function {=} Function
Summary: Specifies the name of freestream pressure versus time function.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Freestream Temperature

Syntax: Freestream Temperature {=} Value
Summary: Specifies a constant freestream temperature. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Freestream Temperature Altitude Function

Syntax: Freestream Temperature Altitude Function {=} Function
Summary: Specifies the freestream temperature versus altitude function for climate condition. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Freestream Temperature Time Function

Syntax: Freestream Temperature Time Function {=} Function
Summary: Specifies the name of freestream temperature versus time function. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Integrated Flux Output

Syntax: Integrated Flux Output VariableName
Summary: Calculate the average flux associated with this flux boundary condition.
Description: This line command specifies that, as a postprocess, the normal flux associated with this boundary condition be integrated over the surface to obtain the total power. This power is then divided by the total area of the surface to obtain the average flux on the surface, and stored in a global variable named “VariableName”. This global variable may then be output to history files, or accessed in user subroutines, etc.

Parameter	Value	Default
VariableName	string	–

Integrated Power Output

Syntax: Integrated Power Output VariableName
Summary: Calculate the total power associated with this flux boundary condition.
Description: This line command specifies that, as a postprocess, the normal flux associated with this boundary condition be integrated over the surface to obtain the total power which is then stored into a global variable named “VariableName”. This global variable may then be output to history files, or accessed in user subroutines, etc.

Parameter	Value	Default
VariableName	string	–

Mach Number

Syntax: Mach Number {=} Value
Summary: Specifies a constant Mach number. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Mach Number Time Function

Syntax: Mach Number Time Function {=} Function
Summary: Specifies the Mach number versus time function for a given flight. The tabulated time is defined relative to the START TIME. Usage: ECKERT CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Function	string	–

Reference Density

Syntax: Reference Density {=} Value
Summary: Specifies the reference density value for the density ratio heat transfer coefficient model. Usage: DENSITY RATIO CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	–

Reference Htc

Syntax: Reference Htc {=} Value
Summary: Specifies the reference heat transfer coefficient value for the density ratio heat transfer coefficient model. Usage: DENSITY RATIO CONVECTIVE COEFFICIENT.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	–

Reference Temperature

Syntax: Reference Temperature {=} Value
Summary: Specify a constant reference temperature for this boundary condition.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	–

Reference Temperature Time Function

Syntax: Reference Temperature Time Function {=} FunctionName
Summary: Specify the name of a time-dependent function for the reference temperature for this boundary condition.

Parameter	Value	Default
{=}	{= \| are \| is}	–
FunctionName	string	–

Start Time

Syntax: Start Time {=} Value
Summary: Specifies the analysis time at which the flight begins and the AERO heat flux will be applied.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	0.0

Stop Time

Syntax: Stop Time {=} Value
Summary: Specifies the analysis time at which the flight ends and the AERO heat flux BC becomes inactive.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	none

Uq Flux Multiplier

Syntax: Uq Flux Multiplier {=} Value
Summary: Specify constant scaling of the convective flux.
Description: Intended use of this scaling parameter is primarily for evaluation of model sensitivities.

Parameter	Value	Default
{=}	{= \| are \| is}	–
Value	real	1.0

Use Toggle Block

Syntax: Use Toggle Block ToggleName [ {on} ElementBlockList… ]
Summary: Specification for toggling entities in the computational model based on Toggle Block parameters. When used at the region level, the list of element blocks to be toggled must be provided. Otherwise a listing of entities is not needed as the Toggle Block will be associated with the command line or the enclosing command block.

Parameter	Value	Default
ToggleName	string	–