Calculate Variables Reference
This chapter details the functions available in the Calculate dialog (accessed via the command on the menu). Formulae, where not trivial, are given for each function. For functions that have equivalent PLOT3D function numbers, the numbers are listed as well. Refer to Selecting a Function for a description of how to use these numbers.
Symbols
The following symbols are used in formulae below. Other symbols are defined in context.
| Symbol | Description |
|---|---|
|
Reference or free-stream quantity. |
g |
Ratio of specific heats, |
r |
Density, mass per unit volume (area in 2D). |
|
Generalized curvilinear coordinate in the I-direction. |
|
Generalized curvilinear coordinate in the J-direction. |
|
Generalized curvilinear coordinate in the K-direction. |
|
Vorticity |
a |
Speed of sound. |
cp |
Specific heat at constant pressure. |
cv |
Specific heat at constant volume. |
M |
Mach number. |
m |
Mass. |
p |
Pressure. |
R |
Specific gas constant |
T |
Temperature. |
|
Velocity vector. |
u |
X-velocity component. |
v |
Y-velocity component. |
w |
Z-velocity component. |
Scalar Grid Quality Functions
I, J, K-aspect Ratio
The ratio of maximum edge length squared to face area:
For a rectangle or square, this simplifies to:
For collapsed faces where the area is zero, the aspect ratio is set to zero.
For Polyhedral zones, the maximum aspect ratio is calculated for all faces of a given cell.
I, J, K-stretch Ratio
The ratio of the length of line segment I2-I3 to segment I1-I2 (or J or K):
or
such that the stretch ratio is always > 1.
If either segment has zero length, the stretch ratio is set to one.
| If you have specified on the dialog that adjacent zones are connected, these stretch ratios will be made continuous across connected zone boundaries provided that the index directions are aligned. |
I, J, K-face Skewness
The ratio of the two face diagonal lengths subtracted from one (the diagonals are ratioed so that this number is always non-negative):
For polyhedral zones, the cell value is the maximum skewness over all cell faces.
Cell Diagonal1 or Diagonal2 Skewness
The ratio of the lengths of two body diagonals subtracted from one (always non-negative). There are four body diagonals. We choose pairs which would be coplanar in an unskewed cell, that is, (i,j,k) → (i+1,j+1,k+1) and (i,j,k+1) → (i+1,j+1,k).
For polygonal zones, the ratio of the lengths will be zero.
IJ, JK, KI, or Max Normals Skewness
The dot product of face unit normals for the two given faces.
The following figure illustrates this for IJ-skewness.
I, J, K, or Min Orthogonality
One minus the absolute value of the dot product of two unit vectors which point in the direction of two adjacent edges of the given face.
I, J, K, or Min Nonplanarity
Two triangles are formed with the four nodes of the face, and the dot product of the two unit normals of those triangles is subtracted from one.
For polyhedral zones, the max over all cell faces is found.
Jacobian
For ordered zones, the Jacobian is calculated with the standard formula.
The subscripts above represent partial derivatives, which are approximated with finite differences.
For finite element zones, Tecplot 360 approximates the Jacobian by inverting the average areas or volumes of the grid cells surrounding each node, 1/A or 1/V.
If the denominator of the above formula is zero (ordered zones), or all cells surrounding a node have zero area (finite element zones), the Jacobian is set to zero.
Cell Volume
For ordered zones, the cell volume for a particular node (I, J, K) is the volume of the cell between nodes (I, J, K) and (I+1, J+1, K+1). In 2D, this function becomes cell area. Nodes on the IMax, JMax, and KMax boundaries are assigned the same value as the nodes at IMax-1, JMax-1, and KMax-1 respectively.
For finite element zones, the cell volume for a node is the average volume (area in 2D) of all cells of which that node is a part.
Scalar Flow Variables
Density
The mass per unit volume of the fluid:
PLOT3D function numbers: 100 (not normalized), or 101 (normalized).
Pressure Coefficient
PLOT3D function number: 114 (not normalized). There is no function number for normalized pressure coefficient, since reference value normalization is not possible (the free-stream pressure coefficient is zero).
Stagnation Pressure Coefficient
PLOT3D function number: 115 (not normalized). As above, there is no function number for normalized stagnation pressure coefficient.
Pitot Pressure
Equals stagnation pressure for subsonic/incompressible flow. For supersonic flow:
PLOT3D function number: 116 (not normalized).
Pitot Pressure Ratio
The pitot pressure divided by the free-stream pressure. PLOT3D function number: 117 (not normalized).
Stagnation Enthalpy
per unit mass:
PLOT3D function numbers: 132 (not normalized), or 133 (normalized).
Stagnation Energy
per unit mass:
PLOT3D function numbers: 142 (not normalized), or 143 (normalized).
Stagnation Energy per Unit Volume
Stagnation energy multiplied by density. PLOT3D function number: 163 (not normalized).
Kinetic Energy
Per unit mass, one-half the square of the velocity magnitude.
PLOT3D function numbers: 144 (not normalized), or 145 (normalized).
Velocity Components U, V, or W
The scalar velocity components. PLOT3D function numbers: 150 (u, not normalized), 151 (v, not normalized), or 152 (w, not normalized).
Velocity Magnitude
The 2-norm of the velocity vector components:
PLOT3D function number: 153 (not normalized).
Mach Number
The flow speed divided by the local speed of sound, for compressible flow:
PLOT3D function number: 154 (not normalized).
Cross Flow Velocity
This presumes that free-stream velocity is purely in the X-direction:
PLOT3D function number: 156 (not normalized).
Equivalent Potential Velocity Ratio
The ratio of velocity magnitude to the potential velocity, as calculated with the incompressible Bernoulli equation. Refer to previous sections for definitions of and .
PLOT3D function number: 159 (not normalized).
X, Y, Z-momentum Component
Per unit volume, the product of density and the scalar velocity components.
PLOT3D function numbers: 160 (X-momentum, not normalized), 161 (Y-momentum, not normalized), 162 (Z-momentum, not normalized).
X-, Y-, Z-Vorticity
PLOT3D function numbers: 180 (X-Vorticity, not normalized), 181 (Y-vorticity, not normalized), 182 (Z-vorticity, not normalized).
Q Criterion
Where and S are the anti-symmetric and symmetric components of the velocity gradient tensor:
and the norms for the tensors are the Frobenius norms—the square root of the sum of the squares of all tensor elements.
Filtered Relative Helicity
as calculated above, but set to zero when:
PLOT3D function number: 188 (not normalized).
Filtered Shock
Shock, as shown above, but set to zero when the magnitude of the pressure gradient
PLOT3D function number: 191 (not normalized).
X, Y, Z-density Gradient
PLOT3D function numbers: 194 (X-density Gradient, not normalized), 195 (Y-density Gradient, not normalized), 196 (Z-density Gradient, not normalized).
Sutherland’s Law
Sutherland’s Law is a method of estimating the viscosity of a fluid from its temperature. The formula is:
For the constants, Tecplot 360 uses the meters/kilograms/seconds values for air,
and C2 = 110.4 K. Unlike other functions, this function is units-specific. Tecplot 360 uses the meters/kilograms/seconds units for this calculation, so the input temperature (data set variable) must be in Kelvin. The resulting viscosity will be in units of: kg /m s.
Vector Flow Variables
The Velocity Gradient Tensor
In addition to the scalar and vector variables listed in the previous sections, Tecplot 360 can calculate one tensor variable, the velocity gradient:
Each component in the tensor is stored as a separate variable in the dataset. The names indicate which component they represent, such as dUdX, dUdY and so on.