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(b4wind User's Guide) (Converting Files) (Computing Initial Conditions) (Interpolation) (Computing the Reynolds Number) (Preparing NPARC or XAIR Namelists) (Installation and Execution) (Conclusions) (Computing the Mach Number From the Area Ratio) (Trilinear Interpolation)

Computing Initial Conditions

If you choose the "Compute initial conditions" option from the main b4wind menu, then the first step is to a read a grid file in PLOT3D, NPARC or Common File format. The GUI leads you through the reading of the grid file which it writes to a temporary file. After the grid file has been read a panel will be displayed to set dimensional units and reference values.

The second step is to compute the initial conditions, choosing from the three options:

Different options can be used for different zones. The details of the options are explained in the following sections. Note that the computations are not done immediately. The GUI only gathers the instructions for computing the flow in each zone. When all of the instructions for computing the flow in all of the zones have been compiled, then a button will appear to go ahead and compute the flow in all of the zones. The solution will then be computed and written to a temporary file. This completes the second step.

The third and final step is to write the solution as a PLOT3D, NPARC or Common file. Again, the GUI leads you through this procedure. The solution is read from the temporary file and written to the specified file using the same programs that were described in the section on Converting Files. If a Common File is written then all of the dimensional units and reference values are known and specified in the file.

Dimensional Units and Reference Values

After the first step of computing initial conditions (reading a grid file) has been completed, a panel titled "Freestream and Reference Values" will be displayed. This panel can be used to set the dimensional units and the reference values used to nondimensionalize the flow equations. If you write the initial conditions solution to a Common File, then these values are stored in the Common File. If you write it to a PLOT3D file, then Mach, Alpha and Re are written with the solution. If you write it to a NPARC file, then Gamma is written.

First the parameters on the panel will be defined.

    L   The reference length specified in the units specified in the radio boxes [Length]
Gamma The ratio of specific heats [dimensionless]
Mach The magnitude of the freestream velocity divided by the speed of sound [dimensionless]
Alpha The freestream angle of attack [degrees]
Beta The freestream angle of yaw [degrees]
Mw The molecular weight [dimensionless]
Rg The gas constant [(Force × Length) / (Mass × Temperature)]
g0 A conversion factor which is set automatically when you specify the units [(Mass × Length) / (Force × Time2)]
UGC The universal gas constant [(Force × Length) / (Mass × Temperature)]
rhoRef The reference density [Mass / Length3]
aRef The reference speed of sound [Length / Time]
pRef The reference pressure [Force / Length2]
TRef The reference temperature [Temperature]
u The x-component of the freestream velocity [Length / Time]
v The y-component of the freestream velocity [Length / Time]
w The z-component of the freestream velocity [Length / Time]
V The magnitude of the freestream velocity [Length / Time]
ERef The reference for the energy per unit volume [Force / Length2]
MuRef The reference viscosity [Force × Time / Length2]
TR A reference temperature used in Sutherland's viscosity formula [Temperature]
TSuth The temperature constant used in Sutherland's viscosity formula [Temperature]
MuR The viscosity at temperature TR [Force Time / Length2]

The components of the freestream velocity, the magnitude and the angles of attack and yaw are related by

u = V cos(Beta) cos(Alpha)
v = V cos(Beta) sin(Alpha)
w = V sin(Beta)

The gas constants and molecular weight are related by

Mw Rg = UGC

The reference pressure and temperature can be set to the freestream static or total values or can be set arbitrarily. When you enter a static value, the total value is computed and vice versa. They are related by the isentropic conditions

TTotal = TStatic (1.0 + 0.5 (Gamma−1.0) Mach2)

pTotal = pStatic (TTotal / TStatic)Gamma/(Gamma−1.0)

Note that the nondimensional pressure is obtained by dividing the dimensional pressure by (Gamma pRef). All of the other quantities are nondimensionalized by dividing the dimensional quantity by the reference value. Other reference values are computed by

aRef = sqrt (g0 Gamma Rg TRef)

rhoRef = pRef / (Rg TRef)

ERef = pRef

The Reynolds Number is computed by

Re = (rhoRef aRef L) / (g0 MuRef)

where MuRef is given by the Sutherland Viscosity Formula

MuRef = MuR [(TR + TSuth) / (TRef + TSuth)] (TRef / TR)1.5

Note at the present, in this panel, there is no way to specify TR and TSuth. This should be corrected. Presently it uses the values shown below to compute MuRef.

Initially the values of the various parameters are set to the values of air at standard (static) temperature and (static) pressure at a Mach Number of 0.5.

    L  =  1.0 ft
g0  =  32.174 ft lbm / lbf sec2
Gamma  =  1.4
Mw  =  28.966
Rg  =  53.3671 ft lbf / lbm °R
pRef  =  14.7 lbf / in2
TRef  =  491.67 °R = 32.0 °F
TR  =  491.67 °R
TSuth  =  198.6 °R
MuR  =  3.584×10−7 lbf sec / ft2
Mach  =  0.5
Alpha  =  0.0
Beta  =  0.0

After you click the Ok button on the "Freestream and Reference Values" panel, you are ready to begin the second step - computing the initial conditions. A panel labeled "Compute Initial Conditions" will come up.

Uniform Flow

In the "Compute Initial Conditions" panel, to set initial conditions to uniform flow click the radio button labeled "Uniform flow" and then click the Next button. Then will be displayed the "Uniform Flow" panel. There you can specify Mach number, Alpha, Beta, and either the static or total pressure and temperature. Note that Gamma is assumed to be the same for all zones and was specified in the "Freestream and Reference Values" panel. This is enough to specify a flow field. Since Gamma and Mach number are known, given the static pressure, the total pressure can be computed from the isentropic formulas and vice versa. The same is true for the temperature. The presures are nondimensionalized by dividing them by (Gamma pRef), and the temperatures are nondimensionalized by dividing them by TRef. Everything that follows is in nondimensional form. The isentropic relations are

TTotal = T (1.0 + 0.5 (Gamma−1.0) Mach2)

pTotal = p (TTotal/T)(Gamma/(Gamma−1.0))

Given the static pressure and temperature, p and T

a = sqrt (T)

V = Mach a

u = V cos(Beta) cos(Alpha)
v = V cos(Beta) sin(Alpha)
w = V sin(Beta)

ρ = Gamma p / T

E = p / (Gamma−1.0) + 0.5 ρV2

Thus are derived the flow variables ρ, ρu, ρv, ρw and E (energy per unit volume or ρ times the stagnation internal energy per unit mass, i.e. ρe0).

Initial Conditions by Interpolation

In the "Compute Initial Conditions" panel, to compute the initial conditions by interpolation, click the radio button labeled "Static pressure or Mach number found by interpolation" and then click the Next button. The "Interpolated Flow" panel will appear. Here supply Alpha, Beta, pTotal and TTotal. You can use the radio buttons to choose whether to interpolate on the static pressure or Mach number and use the other set of radio buttons to choose which index to interpolate with respect to (wrt). Then enter the specified parameter (static pressure or Mach number) at the first and last specified index.

As example, assume the specified index is I and that the specified parameter is M1 at I = 1 and M2 at I = IMAX. Then set

DMDI = (M2M1) / (IMAX1)

then for each I

M = M1 + (I1) DMDI

An exact analogy holds if the specified parameter were static pressure or for another specified index.

Note that the pressures are nondimensionalized by dividing by (Gamma pRef) and TTotal by dividing by TRef all of which are known. If Mach number is the specified parameter, then the static pressure and static temperature can be computed from the isentropic relations and all of the other flow variables can be computed exactly the same way as for uniform flow. If static pressure is the specified parameter, then

(TTotal / TStatic) = (pTotal / pStatic)(Gamma−1)/Gamma

Since everything is known on the right hand side, the ratio on the left hand side can be computed; then

Mach = sqrt (2.0 ((TTotal/TStatic) − 1.0) / (Gamma−1))

Knowing the Mach number all of the flow variables can be computed as before. All of the flow variables are uniform for the specified index.

One-Dimensional Flow

In the "Compute Initial Conditions" panel, to compute initial conditions by one-dimensional flow equations, click the radio button labeled "Mach number is found by area ratios" and then click the Next button. The "One-Dimensional Flow" panel will appear. Here the only numbers required are the total pressure and temperature and, optionally, a throat area and a limit to the Mach number. The rest of the radio buttons are used to specify the valid options. The flow can be along the direction of any index. The cross-sectional areas were computed in all three directions when the grid was read in. The direction of the flow can be that of increasing index or decreasing index. Trying to be general, you can specify which of the physical coordinates changes most with the specified index.

The following options are interrelated so if something is unclear, keep reading. Maybe your question will be answered. Important! Click which zones you are defining before setting the rest of the options; otherwise the areas will be incorrect and it may become confused.

The Global and Local options have to do with throat area selection and are only valid in the Automatic mode. If Global is chosen then the throat area is determined from all of the zones. If Local is chosen, then the throat area is determined only from the selected zones. The throat area thus determined is displayed. If it does not look right, then double check all of the options.

You can choose either Automatic or Specified. If you choose Automatic then the throat area will be determined as described in the preceding paragraph. If Specified is chosen then the entry for the throat area becomes active and you can enter the throat area. The specified area must must be less than the Minimum Area. When the throat area is specified the solution will be either all subsonic or all supersonic.

The Subsonic and Supersonic options are valid only in the Specified mode. When you specify a throat area, it is not known whether the throat is downstream or upstream of the zone. If it is downstream, then the flow will be subsonic. If it is upstream, then the flow will be supersonic. Therefore you can specify whether you want subsonic or supersonic flow.

After all of the above has been specified, for each value of the specified index (I, J or K), in each of the specified zones, the cross-sectional area can be computed. Then the ratio of the cross-sectional area and the throat area can be computed. From this ratio the Mach number can be computed. So now, Mach, pTotal and TTotal are known. This is all that is needed to compute the flow at the particular index. This computation is exactly the same as that presented for uniform flow with Alpha and Beta both zero.

Computation of Cross-Sectional Area

Very briefly, the formulas for cross-sectional area are as follows:

For 3D, let yL and zL be the coordinates along the outer boundary with L being numbered from 1 to n. The cross-sectional area is computed by

A = 0.5 ∑ (yL (zL+1zL−1)

where the Sum is from L = 1 to L = n and where z0 = zn and zn+1 = z1.

For 2D, with i being the designated index and x varying most with i:

A = yi,jmaxyi,1

For axisymmetric, with i being the designated index and x varying most with i:

A = π (yi,jmaxyi,1)2

which is seen to be the cross-sectional area of an annular region if there is a center body (yi,1 not equal 0). The axis of symmetry is assumed to be the x axis (in this case).

Last updated 2 May 2003