construction of v700_5e.dat.txt

This benchmark dataset is v700_5e.dat.txt. To build the dataset the user now provides input to the four data blocks as described below:

Data Block ONE - Geometry, PDE's, Models, Units, and Fluid

###1

'title of dataset'
v700_5e.dat D&B turbine blade, s side at Re,d, ml but no curv, Ts(x)

• title - to appear in all output files

This is a descriptive user-written alphanumeric statement of 80 characters maximum

###2

kgeom	neq	kstart	mode	ktmu	ktmtr	ktme
1	2	7	1	5	5	2

• kgeom - flag for the type of geometry

=1   is a flat plate or planar geometry, and it the required choice to model a turbine blade - the influence of the turbine blade's surface curvature will be supplied through the pressure gradient term (specification of the free stream velocity distribution) and to some extent through modifications to the mixing-length turbulence model.

The governing boundary layer equations only contain transverse curvature terms, because they are formulated in axisymmetric coordinates (see overview: transport equations section of this website), and they reduce directly to planar coordinates by setting the transverse radius of curvature rw variable to unity. The transport equations do not contain flow-direction curvature terms.

Clearly, turbine blade geometries possess surface curvature, which causes the approaching flow streamlines to be displaced and to bend or curve, which in turn causes a radial pressure distribution along the blade surface (a direct result of curved streamlines) and an axial pressure distribution along the blade surface as the flow accelerates and decelerates (a Bernoulli-like effect). Because there are no flow-direction curvature terms the radial pressure distribution over the blade surfaces is ignored in TEXSTAN. The axial pressure distribution over the blade becomes a boundary condition, a consequence of the boundary layer approximations (see the definition of external flow). In TEXSTAN the boundary condition is given by specification of the free stream velocity distribution, obtained either from solving the Euler equations for the blade or by experimental measurement of the blade surface pressure distribution, and TEXSTAN converts this distribution into an axial pressure gradient distribution.

For turbulent flows, one of the mixing-length model choices contains a curvature model term (see modeling: curvature section of this website). At the current time there are no curvature terms for the higher-order turbulence models.

note: TEXSTAN computes the flow for planar geometries on a per unit width basis, so the blade width (root to tip or span) is not be a required input variable. Integration of the heat flux to determine total heat transfer or surface shear stress to determine total shear drag will require multiplying the integral by the value of the blade width.

• neq - flag for the number of transport equations to be solved

=2   means that TEXSTAN will solve the momentum equation and one other equation.

For turbine blade heat transfer with variable properties and viscous work or dissipation (compressible flow) TEXSTAN must solve the stagnation enthalpy equation. Note that TEXSTAN does not have to solve the conservation of mass equation because of the stream-function transformation of the governing equations (see overview: transport equations section of this website).

The turbine blade problem can be computed as a pure momentum problem (neq =1) to determine a displacement thickness distribution for incorporation into an Euler solution. However, the effects of viscous heating and variable properties are quite important for most turbine blade calculations, so a pure momentum solution has limited usefulness.

For turbine blade heat transfer, the flag that will define that the "other transport equation" is the variable jsor. For the turbine blade problem the user will choose jsor =3 and that will tell TEXSTAN that the second transport equation is the stagnation enthalpy equation with inclusion of viscous work (to reflect viscous heating and viscous dissipation). When higher order turbulence modeling is being used, the neq value will increase and there will be a specific jsor value for each turbulence model equation (for example the k equation and the ε equation).

• kstart - flag to specify the type of initial profiles and whether they will be TEXSTAN-generated or input from tables

=7   instructs TEXSTAN to generate Falkner-Skan stagnation point laminar initial profiles for velocity and temperature - a reasonable model for the leading edge region of the turbine blade and valid with any turbulence model choice.

The Falkner-Skan profiles are applicable to flow over a flat plate (Blasius, m=0), and flow in a 2-D stagnation region (m=1), applicable to 2-D stagnation-point flow, flow over a cylinder, or flow at the leading edge region of a turbine blade. The laminar profiles also serve to start transitional flows. The Falkner-Skan starting solution is described in detail in external flows: initial profiles. It is a compressible solution, valid in the range of fluids for gases and light liquids, and the solution is extended within TEXSTAN to compute initial starting profiles for the k equation and the ε equation when needed.

• mode - flag to tell TEXSTAN whether to compute turbulent viscosity at the start of the calculations

=1   means the flow will start laminar, which is correct for the leading-edge region of a turbine blade if the user is choosing a mixing-length model or one-equation model of turbulence.

If the turbulence model is changed to consider the k equation and the ε equation, then the value of mode will be set =2, and the flow will begin turbulent, but the turbulence effects will be controlled by a transition model to minimize its initial effects (depending on the level of the approach turbulence intensity) and control its build up (see modeling: transition section of this website). At the present time there is no mechanism to adapt the mixing length model to the effects of high free stream turbulence (for example, a turbulized laminar stagnation layer).

• ktmu - the flag to tell TEXSTAN what to use for a momentum equation turbulence model

=5   is a mixing-length model of turbulence that has been adapted to pressure-gradient flows.

The turbine blade dataset is designed to permit the user to investigate a number of turbulence models (see the modeling section of this website). For example, a model for the effect of blade curvature can be added to the mixing-length model by setting kbfor=5. A one-equation model can be considered by setting ktmu=11, and various two-equation models can be investigated as well with appropriate changes to the flags and variables.

• ktmtr - the flag to tell TEXSTAN what to use for a transition turbulence model.

=5   is a turbulence transition model that works well with the mixing-length model, although it has had limited testing with complex pressure gradient distributions.

There are four transition models that work with the mixing length turbulence model as described in the modeling: transition section of this website). The abrupt model permits the user to choose the transition point, and the other three models have some level of calibration with experimental data. However, transition is influenced my numerous variables, especially for turbine blading, and the TEXSTAN models are at best approximate. The choice of the two-equation Lam-Bremhorst turbulence model along with its transition model may be more accurate.

• ktme - flag to tell TEXSTAN what to use for an energy equation turbulence model

=2   is a constant turbulent Prandtl number, recommended for complex pressure-gradient flows or transitional flows.

The use of a constant turbulent Prandtl number seems to be successful for complex flows such as a turbine blade flow. The choice of a variable turbulent Prandtl number was developed for flows over flat plates with pressure gradients and transpiration, and it is quite successful with these flows. It does not work well with transitional flows.

###3

kbfor	jsor(1)	jsor(2)	jsor(3)	jsor(4)	jsor(5)
1	3	11	21

• kbfor - the flag to tell TEXSTAN what body forces to include in the momentum equation source term

=1   means only the pressure gradient, dp/dx, will be calculated as a momentum source term. A turbine flow is a high Reynolds number flow and not influenced by body-force work from free convective body forces.

The kbfor=1 option is the required choice for almost all internal and external forced convection TEXSTAN problems. This choice includes the flat plate problem (for example the Blasius problem where u_∞= constant, and the computed pressure gradient is zero for all x).

The kbfor=5 option causes TEXSTAN to process the aux3(m) array to introduce a curvature modification to the mixing length formulation (see modeling: curvature section of this website).
note: the turbine blade dataset contains blade curvature data in the aux3(m) array to permit the user to test this effect - the array is not used unless kbfor is set =5 and the turbulence model is the mixing-length turbulence model.

• jsor(j) - the flag to tell TEXSTAN what each diffusion equation will be and which set of source terms to include when formulating the general source for this diffusion equation

jsor(j=1)=3 along with kfluid>1 (variable properties) tells TEXSTAN the first diffusion equation after the momentum equation will be a stagnation enthalpy equation and viscous work must be calculated for this equation (related to viscous dissipation).

jsor(j=2)=11 is specified, even though neq=2 means the variable will not be read as part of the TEXSTAN input. This is one of the "extra variables" described in the Input Dataset part of the turbine blade section so that the dataset can be easily switched from mixing-length to a two-equation turbulence model. The variable has a value =11 tells TEXSTAN the third transport equation to be solved will be a turbulence kinetic energy equation, and the variable ktmu will tell TEXSTAN which turbulence kinetic energy equation to use.

jsor(j=3)=21 is also specified, and this is another "extra variable". The variable has a value =21 tells TEXSTAN the fourth transport equation to be solved would be a turbulence dissipation equation.

jsor(j=4) and jsor(j=5) are left blank - these variables are not needed and will not be read as a part of the TEXSTAN input (choosing the mixing-length model means neq=2 and only one jsor value needed; choosing the two-equation model means neq=4 and only three jsor values needed).

The use of neq and jsor reflect the architecture of the numerical formulation within TEXSTAN. This is discussed in some detail in the Generalized Transport Equation part of overview: transport equations section of this website. The neq variable controls how many differential transport equations will be solved (at this time there are a significant number that are programmed into TEXSTAN, including momentum equation, temperature equation, stagnation enthalpy equation, mass transport equation for a binary mixture, five turbulence kinetic energy equations, and four turbulence dissipation equations). If neq=1 only the momentum equation will be solved, and when neq=4 the momentum and three other transport equations will be solved. The =4 value is chosen, for example, when the user wants to solve (say) the momentum equation, the stagnation enthalpy equation, a particular turbulence kinetic energy equation, and its corresponding turbulence dissipation equation. It is the jsor variable that identifies all the equations other than the momentum equation (which is solved by definition) that will be solved by TEXSTAN.

The fact that jsor(j=1) has a value in the range between 1-8 tells TEXSTAN the second transport equation to be solved (momentum being the first) will be an energy equation. The energy equation will be considered a temperature equation if the properties are assumed constant (kfluid=1), and it will be considered a stagnation enthalpy equation if the properties are assumed variable (kfluid>1). Regardless of whether it is a temperature or stagnation enthalpy equation, the general form of the equation is convection = diffusion +/- sources, i.e. convective transport of temperature or stagnation enthalpy is balanced by its diffusive transport (laminar + turbulent) and by all possible sources and/or sinks (positive and negative sources) for the generation or destruction of temperature or stagnation enthalpy while it is in transport. An example of a source would be viscous dissipation (for the temperature equation) or viscous work (for the stagnation enthalpy equation). The convection - diffusion - source/sink formulation is described in detail in the Generalized Transport Equation part of overview: transport equations section of this website.

###4

kfluid	kunits
2	1

• kfluid - the flag to tell TEXSTAN whether to use constant or variable properties

=2   is the choice for variable properties, appropriate to a gas turbine flow.

The various choices for variable property tables within TEXSTAN are described in overview: fluid properties. This choice (=2) is for dry air at low pressure from CHMT, which means the effects of pressure on the thermophysical properties is ignored, except for density. The air property routine kfluid=13 is also a good choice, and the differences in surface friction and heat transfer is a second order effect. With a gas turbine, air is an approximation because the working fluid is really products of combustion. The variable property routine kfluid=14 is a generic perfect gas model, and it can also be used to model air or products of combustion. It has the advantage of specifying the molecular weight of the gas, but its disadvantage is that the input variables of fluid specific heat ratio and Prandtl number are treated as constant input values (the perfect gas model), rather than being functions of temperature.

• kunits - the flag to tell TEXSTAN whether to use SI (metric) units or U.S. Customary (English/British) units for all physical variables

=1   specifies SI for all the units.

All dimensional variables in TEXSTAN follow a units convention, as described in overview: units. TEXSTAN is formulated to permit the user either to choose U.S. Customary units or SI units, and all the benchmark datasets are presented in SI units.

###5

po	rhoc	viscoc	amolwt	gam/cp
290000.0	0.0	0.0	0.0	0.0

• po - the fluid pressure at the start of computations, x = xstart

=290000.0   in units of N/m² or Pa - the fluid pressure is interpreted as stagnation (total) pressure because of the variable properties assumption, kfluid>1.

The only direct use of pressure in TEXSTAN is for computing fluid density when kfluid>1 (variable properties), for both external and internal flows. The only other use of pressure is with internal flows where pressure drop along the flow direction is computed, but this is an output variable and not used within the calculations.

• rhoc - the fluid mass density

=0.0   

If variable properties are being used, this input variable is set equal to zero. The fluid mass density for kfluid>1 (variable properties) is computed within TEXSTAN using the ideal gas model for all gases, and it is a table lookup or correlation for all liquids. The variable rhoc is only used for kfluid=1 (constant properties). WARNING: the input file to TEXSTAN requires these variables to be zero values, rather than be left blank.

• viscoc - the fluid dynamic viscosity

=0.0   

If variable properties are being used, this input variable is set equal to zero. The fluid dynamic viscosity for kfluid>1 (variable properties) is either computed within TEXSTAN using the Sutherland viscosity correlation or a table lookup for all gases, and it is a table lookup or correlation for all liquids. The variable viscoc is only used for kfluid=1 (constant properties).

• amolwt - the fluid molecular weight

=0.00   

This variable is only used for the variable properties model kfluid=14. Otherwise this input variable is set equal to zero.

• gam/cp - the ratio of the fluid specific heats (or) the fluid specific heat

=0.0  

This variable is only used for the variable properties model kfluid=14 and for constant properties (kfluid=1). The gam/cp variable is the only TEXSTAN input variable that is confusing in the sense it has a double interpretation. For kfluid=14 (variable properties) it is gamma, the ratio of specific heats, and for kfluid=1 (constant properties) it is the cp, the fluid specific heat. Otherwise this input variable is set equal to zero.

###6

prc(1)	prc(2)	prc(3)	prc(4)	prc(5)
0.0	1.0	1.0

• prc(j) - generalized transport equation C variable for each diffusion equation

prc(j=1)=0.0 because the j=1 transport equation is a stagnation enthalpy equation, which requires variable properties, and TEXSTAN will look up the value for the laminar Prandtl number. The variable prc(1) would only be specified if kfluid=1 (constant properties).

prc(j=2)=1.0 is specified, even though neq=2 means the variable will not be read as part of the TEXSTAN input. This is one of the "extra variables" described in the Input Dataset part of the turbine blade section so that the dataset can be easily switched from mixing-length to a two-equation turbulence model. The variable has a value =1.0 which is required for the laminar C variable of the turbulence kinetic energy equation.

prc(j=3)=1.0 is also specified, and this is another "extra variable". The variable also has a value =1.0 which is required for the laminar C variable of the turbulence dissipation equation.

prc(j=4) and prc(j=5) are left blank - these variables are not needed and will not be read as a part of the TEXSTAN input (choosing the mixing-length model means neq=2 and only one prc value needed; choosing the two-equation model means neq=4 and only three prc values needed).

From the introduction to the variable jsor(j) we saw that there can be up to five "j" variables: jsor(j=1) to jsor(j=5). For neq=2 there will be (neq - 1) values of the jsor(j) variables = one "j" variable required. For neq=4 there will be (neq-1) values of the jsor(j) variables = three "j" variables required.

In the Generalized Transport Equation part of overview: transport equations section of this website, the C variable in the diffusion term of the general transport equation (convection = diffusion +/- sources) represents the combined laminar and turbulent diffusional transport. Recall (again from this section of the website) the turbulent diffusional transport is a result of assuming mean field closure.

The prc(j) variables are the laminar part of the general transport equation C variable. For the energy equation, It is the C=Pr (the laminar Prandtl number of the fluid). For the mass transfer equation it is the C=Sc (the laminar Schmidt number), and for turbulence kinetic energy and turbulence dissipation equations C=1.0 is required.

Data Block TWO - Boundary Conditions

###7 (and) ###8

nxbc(I)	jbc(I,1)	jbc(I,2)	jbc(I,3)	jbc(I,4)	jbc(I,5)
19	1	1	1

nxbc(E)	jbc(E,1)	jbc(E,2)	jbc(E,3)	jbc(E,4)	jbc(E,5)
19	1	1	1

• nxbc(I) - number of boundary condition locations, I-surface

=19   is the number of experimental data points for the suction-side free stream velocity distribution. This value needs to be on the order of 20 or larger when the free stream velocity varies in the spanwise direction, because the data are spline-fit and differentiated to form the free stream pressure gradient distribution, a direct source term in the momentum equation. This is further discussed in the presentation of the free stream velocity variable, ubE(m).

• nxbc(E)=nxbc(I) - number of boundary condition locations, E-surface

=19    which must be identical to nxbc(I)

• jbc(I,j) - the flag to tell TEXSTAN type of boundary condition at the I-surface (Dirichlet or level, Neumann or flux, or symmetry line) for each diffusion equation

jbc(I,j=1)=1   tells TEXSTAN that the boundary condition along the I-surface for the first diffusion equation (the energy equation) will be mathematically a Dirichlet or level type, which implies a surface temperature distribution along the I-surface (if it had been =2, it would have been mathematically a Neumann or flux type, which would imply a surface heat flux distribution along the I-surface).

jbc(I,j=2)=1 is specified, even though neq=2 means the variable will not be read as part of the TEXSTAN input. This is one of the "extra variables" described in the Input Dataset part of the turbine blade section so that the dataset can be easily switched from mixing-length to a two-equation turbulence model. The variable has a value =1 which is required for the turbulence kinetic energy equation, regardless of choice of two-equation model.

jbc(I,j=3)=1 is also specified, and this is another "extra variable". The variable also has a value =1 which is required for the turbulence dissipation equation, regardless of choice of two-equation model.

jbc(I,j=4) and jbc(I,j=5) are left blank - these variables are not needed and will not be read as a part of the TEXSTAN input (choosing the mixing-length model means neq=2 and only one jbc value needed; choosing the two-equation model means neq=4 and only three jbc values needed).

• jbc(E,j) - the flag to tell TEXSTAN type of boundary condition at the E-surface (Dirichlet or level, Neumann or flux, or symmetry line) for each diffusion equation

jbc(E,j=1)=1   tells TEXSTAN that the boundary condition along the E-surface for the first diffusion equation (the energy equation) will be mathematically a Dirichlet or level type, which translates into a free stream temperature. Note that TEXSTAN will use the input variable tstag to specify the free stream temperature.

jbc(E,j=2)=1 is specified, even though neq=2 means the variable will not be read as part of the TEXSTAN input. This is one of the "extra variables" described in the Input Dataset part of the turbine blade section so that the dataset can be easily switched from mixing-length to a two-equation turbulence model. The variable has a value =1 which is required for the turbulence kinetic energy equation, regardless of choice of two-equation model.

jbc(E,j=3)=1 is also specified, and this is another "extra variable". The variable also has a value =1 which is required for the turbulence dissipation equation, regardless of choice of two-equation model.

jbc(E,j=4) and jbc(E,j=5) are left blank - these variables are not needed and will not be read as a part of the TEXSTAN input (choosing the mixing-length model means neq=2 and only one jbc value needed; choosing the two-equation model means neq=4 and only three jbc values needed).

###9

x(m)	rw(m)	aux1(m)	aux2(m)	aux3(m)
0.0000000	1.0000	0.0000	0.0000	1.2789E-02
0.0005000	1.0000	0.0000	0.0000	1.2789E-02
0.0010000	1.0000	0.0000	0.0000	1.2789E-02
0.0020000	1.0000	0.0000	0.0000	1.2789E-02
0.0040000	1.0000	0.0000	0.0000	1.0208E-02
0.0088000	1.0000	0.0000	0.0000	6.7012E-03
0.0126100	1.0000	0.0000	0.0000	8.1462E-03
0.01636900	1.0000	0.0000	0.0000	1.3176E-02
0.0202000	1.0000	0.0000	0.0000	2.1763E-02
0.0252000	1.0000	0.0000	0.0000	3.8210E-02
0.0308900	1.0000	0.0000	0.0000	6.3793E-02
0.0340500	1.0000	0.0000	0.0000	8.1033E-02
0.0378300	1.0000	0.0000	0.0000	1.0439E-01
0.0416100	1.0000	0.0000	0.0000	1.3063E-01
0.0454000	1.0000	0.0000	0.0000	1.5973E-01
0.0498000	1.0000	0.0000	0.0000	1.9687E-01
0.0542000	1.0000	0.0000	0.0000	2.3746E-01
0.0599000	1.0000	0.0000	0.0000	2.9500E-01
0.0643000	1.0000	0.0000	0.0000	3.0618E-01

• x(m) - the boundary condition x-location array

As seen in the x(m) array table in units of meters, there are nineteen points, corresponding to nxbc(I)=nxbc(E)=19.

For any flow, including the turbine blade, x is measured intrinsically along the I-surface. For the turbine blade, x=0.0 m is the leading edge, and the x(m=19) value, x=0.0643000 m is the trailing edge location, but this is not required. Note that (x must be is meters, because kunits=1).

• rw(m) - the transverse curvature array that contributes to defining a given geometry

As seen in the rw(m) array table, these values are arbitrarily set =1.0 because the geometry (kgeom=1) is a planar surface. Most important is that the rw(m) transverse radius value be a constant value and =1.0 is convenient. Regardless of what value is chosen, the transverse radius divides out of the axisymmetric-formulated transport equations.

• aux1(m) - an auxiliary array that is used to provide additional input data to TEXSTAN

As seen in the aux1(m) array table, this variable is not required, but it must be filled with zero values because it is part of the structured data input for TEXSTAN.

• aux2(m) - an auxiliary array that is used to provide additional input data to TEXSTAN

As seen in the aux2(m) array table, this variable is not required, but it must be filled with zero values because it is part of the structured data input for TEXSTAN.

• aux3(m) - an auxiliary array that is used to provide additional input data to TEXSTAN

This array is contains values relating to the flow-direction radius of curvature (in meters, because kunits=1), for use in the mixing length model when kbfor=5. The aux3(m) array is always read as part of the TEXSTAN input. This is one of the "extra variables" described in the Input Dataset part of the turbine blade section so that the dataset can be easily switched to permit testing of the curvature modification to the mixing length model of turbulence (see modeling: curvature section of this website)

###10

ubI(m)	am(I,m)	fj(I,1,m)	fj(I,2,m)	fj(I,3,m)	fj(I,4,m)	fj(I,5,m)
ubE(m)	am(E,m)	fj(E,1,m)	fj(E,2,m)	fj(E,3,m)	fj(E,4,m)	fj(E,5,m)
0.00	0.000	289.0	0.0	0.0
0.00	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
59.32	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
121.72	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
211.97	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
293.26	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
327.80	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
329.25	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
328.11	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
0.00340.27	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
372.46	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
396.43	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
397.21	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
391.79	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
385.34	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
0.00384.09	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
0.00396.93	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
419.73	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
430.89	0.000	0.0	0.0	0.0
0.00	0.000	289.0	0.0	0.0
398.83	0.000	0.0	0.0	0.0

• ubI(m) - the momentum flow-direction velocity boundary condition array at the I-surface

The ubI(m) array for external flows (kgeom=1,2,3) is the no-slip velocity boundary condition at the solid surface, and all values are set =0.00.

• ubE(m) - the momentum flow-direction velocity boundary condition array at the E-surface

The ubE(m) array for external flows (kgeom=1,2,3) is the free stream velocity boundary condition in units of meters per second at the solid surface, obtained from Daniels experimental data for the suction-side free stream velocity distribution. A typical turbine blade free stream velocity distribution will require at a minimum about 20 points to be accurately interpolated, and more points are better. It is imperative the velocity profile be differentially smooth, because TEXSTAN spline-fits the velocity profile, then differentiates it to compute the local pressure gradient via the Bernoulli equation.

Boundary layer computations of turbine blade flows are really quite difficult, as can be seen from maybe 20-30 years of journal articles (as well as conference papers which have very limited distribution) that compare experimental data with simulations. Turbine blade flows are influenced by many variables that reflect their geometry, the imposed turbulence field, three-dimensional effects, transonic and shock-interaction effects, and so forth. One influence that the user can control is the pressure gradient field smoothness that is forced on the boundary layer by way of the free stream velocity distribution. Both Euler solutions and experimental data distributions are never differentially smooth. What TEXSTAN can do is analyze the free stream velocity distribution that the user intends to use as a velocity boundary condition (see overview: bc smoothness section of this website). It is very important that the user work with the velocity boundary condition array, making small adjustments to the free stream velocity values, until the TEXSTAN analysis shows the free stream velocity array produces a smooth pressure gradient distribution.

• am(I,m) - the momentum cross-stream mass flux boundary condition array at the I-surface

The am(I,m) array for external flows (kgeom=1,2,3) is the wall transpiration (cross-stream or y-directed mass flux) at the solid surface, and all values are set =0.000 for impermeable surfaces.

• am(E,m) - the momentum cross-stream ( or y-directed) mass flux boundary condition array at the E-surface

The am(E,m) array for external flows (kgeom=1,2,3) would be the cross-stream (y-directed) mass flux at the free stream, called the entrainment. All values are set =0.0 for any external flow because it can not be prescribed as a boundary condition. Note that it can not be left blank because it is part of the structured data input for TEXSTAN.

• fj(I,j,m) - the diffusion equation boundary condition array at the I-surface for each j at each x(m) location

fj(I,j=1,m) is the I-surface boundary condition for the first diffusion equation. For the turbine blade problem, jsor(j=1) =3 means it is a stagnation enthalpy equation and jbc(I,j=1) =1 means the surface boundary condition for this equation is a Dirichlet (or level) type. For a stagnation enthalpy equation, the surface enthalpy distribution will be required, but TEXSTAN is programmed to permit a surface temperature distribution, and it converts this distribution into an enthalpy distribution for the appropriate value of kfluid. Therefore, from the table entries fj(I,j=1,m) = 289.0 is the surface temperature in degrees Kelvin. If the blade surface was adiabatic, jbc(I,j=1) =2 (Neumann, or surface heat flux distribution), the table entries for fj(I,j=1,m) would be =0.0 (again, technically a Neumann condition is a specification of either a temperature gradient for constant properties or an enthalpy gradient for variable properties, but TEXSTAN is programmed to permit a heat flux distribution and converts it to the appropriate gradient).

fj(I,j=2,m) is the I-surface boundary condition for the second diffusion equation. For the turbine blade problem, jsor(j=2) =11 means it is a turbulence kinetic equation and jbc(I,j=2) =1 means the surface boundary condition for this equation is a Dirichlet (or level) type. In this dataset it is specified, even though neq=2 means the variable will not be read as part of the TEXSTAN input. This is one of the "extra variables" described in the Input Dataset part of the turbine blade section so that the dataset can be easily switched from mixing-length to a two-equation turbulence model. The variable has a value =0.0 which is required for the turbulence kinetic energy equation, regardless of choice of two-equation model.

fj(I,j=3,m) is the I-surface boundary condition for the third diffusion equation. For the turbine blade problem, jsor(j=3) =21 means it is a turbulence dissipation equation and jbc(I,j=3) =1 means the surface boundary condition for this equation is a Dirichlet (or level) type. This is another "extra variable", and the variable has a value =0.0 which is required for the turbulence dissipation equation, regardless of choice of two-equation model.

fj(I,j=4,m) and fj(I,j=5,m) are left blank - these variables are not needed and will not be read as a part of the TEXSTAN input (choosing the mixing-length model means neq=2 and only one fj value needed; choosing the two-equation model means neq=4 and only three fj values needed).

• fj(E,j,m) - the diffusion equation boundary condition array at the E-surface for each j at each x(m) location

fj(E,j=1,m) is the E-surface boundary condition (free stream) for the energy equation, and for both forms of the energy equation (temperature or stagnation enthalpy), the value is set =0.0, and instead it is specified by the input variable tstag. For the temperature form of the energy equation tstag will be interpreted as the free stream static temperature, and for the stagnation enthalpy equation tstag will be the stagnation temperature. Within TEXSTAN the stagnation temperature will be converted to a stagnation enthalpy for the appropriate value of kfluid. For either energy equation formulation, the value of tstag is a fixed value. Note the variable values can not be left blank because they are part of the structured data input for TEXSTAN.

fj(E,j=2,m) and fj(E,j=3,m) are the E-surface boundary conditions (free stream) for the turbulence kinetic energy equation and the turbulence dissipation equation, and the values are set =0.0, and instead they are calculated as a function of x in TEXSTAN based on their approach values. This procedure is described in detail in the boundary conditions part of modeling: two-equation model section of this website. The variables are specified, even though neq=2 means the variables will not be read as part of the TEXSTAN input. They are part of the "extra variables" described in the Input Dataset part of the turbine blade section so that the dataset can be easily switched from mixing-length to a two-equation turbulence model.

fj(E,j=4,m) and fj(E,j=5,m) are left blank - these variables are not needed and will not be read as a part of the TEXSTAN input (choosing the mixing-length model means neq=2 and only one fj value needed; choosing the two-equation model means neq=4 and only three fj values needed).

Data Block THREE - Integration and Print Control

###11

xstart	xend	deltax	fra	enfra
0.0001100	0.064300	0.050	0.010	1.000E-04

• xstart - the location for the start of computations, x = xstart

=0.0001100   in meters. The value of xstart is calculated by first assuming an x-Reynolds number for the start of computations and then back-calculate what x will be needed to achieve this Re. Note that the user will need to interpolate the free stream velocity distribution to make this back-calculation. For the turbine blade problem, the choice for xstart leads to a starting Re of about 125, which guarantees a starting location within about 3-5 degrees of the stagnation point of the blade, where the approximation of a two-dimensional stagnation point solution for the starting profiles is still correct. Beyond about 10-15 degrees the cylinder-in-crossflow solution would deviate from the stagnation-point solution. The low starting Reynolds number equates to a momentum thickness Reynolds number of about 4, which is also sufficient for two-equation modeling. Note that x = xstart = 0.0 is not an option, because of the TEXSTAN von Mises numerical transformation issue, described in the generalized transport equation part of overview: transport equations section of this website.

• xend - the location for the end point of computations, x = xend

=0.064300   in meters. The two typical choices for xend are to let xend = x(nxbc) and to let xend be a defined x-Reynolds number. For the turbine blade problem, the choice for xend is x(nxbc), the trailing edge, which leads to an end-point Re of about 2E+06, and the analysis of the free stream velocity boundary condition data shows the flow remains attached to this point. Note that if the data (or TEXSTAN) indicate a separation location before the trailing edge, a value before separation should be used for xend to prevent data output problems when the computations are shut down due to the inevitable divide-by-zero condition.

• deltax - the nondimensional integration stepsize variable

=0.050    (dimensionless). This is a recommended value for transitional and fully turbulent flows in which there are no abrupt changes in thermal boundary conditions (such as an unheated starting length). The value was determined based on exhaustive testing of zero-pressure gradient fully turbulent benchmark datasets and comparison of the predictions with accepted turbulent friction and heat transfer correlations.

• fra - the nondimensional variable that contributes to limiting integration stepsize

=0.010    (dimensionless). This is a recommended value for laminar, transitional, and turbulent flows. It is part of the set of integration controls for external flows, and it is designed to limit stepsize in the event that the boundary layer grows too fast. The deltax variable makes integration stepsize proportional to the boundary layer thickness, and a strong increase in that thickness leads to an increase in physical integration stepsize, which generally results in a perceived need to further increase entrainment. The entrainment in turn causes the boundary layer to grow, and this potentially negative feedback loop is limited by fra. Typically this variable is activated only for regions of strong adverse pressure gradient on the turbine blade, to try to control the behavior of the numerical results in the event a separation point is encountered.

• enfra - the nondimensional variable that contributes to limiting integration stepsize

=1.000E-04    (dimensionless). This is a recommended value for turbine blade flows that have both transition and pressure gradient effects. It is part of the set of integration controls for external flows, and it is designed to help ensure the derivatives of dependent variables (the variables are identified by the flag kent) will be as close to zero as possible as they approach the edge of their respective 99-percent boundary layer thicknesses. The value was determined by exhaustive testing of transitional benchmark datasets and turbine blade datasets and comparison with experimental results. For laminar and turbulent flows where the free stream velocity is constant, a value on the order of 1.000E-06 is more than sufficient, and for pressure-gradient flows where the boundary layer thicknesses are continuously changing, this is generally too strict.

Note: Numerical simulations always require testing to determine whether the solution is grid and stepsize independent. In TEXSTAN testing is carried out by refining the resolution of the initial profile and by changing deltax and enfra. The use of kstart >0 (which allows TEXSTAN to generate initial profiles that have been well tested for grid spacing) and use of the recommended values for deltax and enfra are highly recommended. Please see the overview: numerical accuracy section of this website for a complete discussion of grid and stepsize independence.

###12

kout	kspace	kdx	kent
9	200	0	1

• kout - the flag to tell TEXSTAN which formatted output routine to use for the output file out.txt

=9    Turbine blade flows show the affects of acceleration, transition, and deceleration, which make the surface friction and heat transfer hard to interpret. However, the surface friction and heat transfer results need to be within an order of magnitude of zero pressure gradient results. A special output file format was developed for transitional flows (a turbine blade flow is a transitional flow) to better assess the output data. This option is kout=9, which compares TEXSTAN-predicted friction and heat transfer with accepted laminar correlations (Blasius) for the laminar portion of the flow and accepted turbulent correlations for the turbulent portion of the flow. Clearly, in the transitional region the ratio of TEXSTAN results to correlations will be significantly different from unity, and for the turbine blade, the ratio will also be affected by pressure gradient and variable property effects.

The output variable choice kout =2 can also print external flow results. Whether the user chooses kout =2 or =9, the resulting output file out.txt is used primarily for an overview of the integration results, and it is not formatted for data analysis. For data analysis, the integration results are reformatted into a series of ftn.txt files for this purpose, and these files will be written when the flag k5 is set > 0.

• kspace - the flag to choose the interval for printing integration steps in the file out.txt

=200    This choice means that every 200 integration steps a line of output will be written. This flag is very useful for controlling the overall size of the file out.txt, especially for printing purposes, and it is in keeping with the paragraph above which states that the file out.txt is for integration overview and not for generating detailed data for plotting purposes.

• kdx - the flag to cause TEXSTAN to bypass the use of the deltax variable for integration stepsize control and instead use interpolated values from the aux1(m) array

=0    This choice means TEXSTAN will use the deltax variable for integration stepsize control and not use the aux1(m) array. The use of kdx=1 and the aux1(m) array is mostly for internal flows or for any flow where there is a step-change in surface thermal boundary condition, where the user would want a fine resolution in output immediately following the step.

• kent - the flag for external flows to tell TEXSTAN which dependent variables to examine for numerical accuracy related to entrainment

=1    This choice means TEXSTAN will monitor the near free stream derivatives of both the momentum (velocity) boundary layer and the energy boundary layer, and changes to entrainment will be based on whichever boundary layer has a derivative that is becoming too large. For gases, kent=1 is especially important because the fluid Prandtl number is < 1, and the thermal boundary layer will be larger than the velocity boundary layer. Choosing kent=0 (entrainment based only on the velocity boundary layer) will result in numerical accuracy problems. For extension of the problem statement to a two-equation turbulence model, a flag value =1 is still recommended, although at times the turbulence kinetic energy equation boundary layer thickness and especially the turbulence dissipation equation boundary layer thickness can be troublesome (this is the reason there are two other kent options).

###13

k1	k2	k3	k4	k5	k6
0	0	0	0	50	0

• k1 - flag is not used

• k2 - flag is not used

• k3 - flag is not used

• k4 - flag is not used

• k5 - flag to choose the interval for printing integration steps in various ftn.txt files

=50    When the flag is set >0 various ftn.txt files will be printed that contain arrays of data for plotting, and the data will be printed at multiples of k5 integration steps (this flag functions exactly the same as the kspace flag for setting the print interval of out.txt). By control of the size of k5 the user can match the requirements of a plotting package (in terms of maximum number of data points for display). Descriptions of the ftn.txt files for external flows and their contents are given in the external flows: output files section of this website.

• k6 flag is not used

###14

k7	k8	k9	k10	k11	k12
0	0	0	22	0	0

• k7 - flag is not used

• k8 - flag is not used

When the flag is set =36, the free stream velocity array will be analyzed, arrays will be printed for plotting to check for smoothness, and TEXSTAN will be stopped, rather than continue execution. It is important to analyze the tex_edge.txt file and adjust the ubE(m) free stream velocity array if required. This is explained in detail in the overview: bc smoothness section of this website.

• k9 - flag is not used

• k10 - flag to control printing of velocity, temperature, and other dependent-variable profiles in a data format for plotting

=22    When the flag is set >0, profiles will be printed according to the choice of k10. For the turbine blade, the choices are =21,22,23, which permit one set of profiles to be printed at a specified x-Reynolds number, momentum thickness Reynolds number, or enthalpy thickness Reynolds number respectively (specified by the input variable bxx). The files go to various ftn.txt files will be printed that contain arrays of data for plotting, and the data will be printed at multiples of k5 integration steps (this flag functions exactly the same as the kspace flag for setting the print interval of out.txt). By control of the size of k5 the user can match the requirements of a plotting package (in terms of maximum number of data points for display). Descriptions of the ftn.txt files for external flows and their contents are given in the external flows: output files sections of this website.

• k11 - flag is not used

• k12 - flag is not used

###15

axx	bxx	cxx	dxx	exx	fxx	gxx
0.000E+00	2000.0	0.000E+00	0.000E+00	0.000E+00	0.9	0.000E+00

• axx - extra variable for turbulence models or free stream velocity model

• bxx - extra variable for turbulence models or free stream velocity model or print control

=2000.0    When the flag k10 is set in the range 21-25, one set of profiles are printed at a specified location, which for k10=22 is a momentum thickness Reynolds number. Thus, for the turbine blade the set of profiles will be printed after transition is completed and the flow on the blade surface is fully turbulent. If ktmu is set to use a two-equation turbulence model, a number of additional ftn.txt files are printed that contain the components of the turbulence model.

• cxx - extra variable for turbulence models or free stream velocity model

• dxx - extra variable for turbulence models or free stream velocity model

• exx - extra variable for turbulence models or print control

• fxx - extra variable for turbulent Prandtl number

• gxx - extra variable for turbulence models or free convection

Data Block FOUR - Initial Conditions

###16

dyi	rate	tstag	vapp	tuapp	epsapp
5.000E-05	0.0900	432.0	146.40	3.0	0.0

• dyi - this variable is used in the construction of the y spacing for the velocity profile, which in turn controls the grid (finite difference mesh) for TEXSTAN

=5.000E-05    This value is recommended for most boundary layer flows. It is explained in considerable detail in the external flows: initial profiles sections of this website.

• rate - this variable is used in the construction of the y spacing for the velocity profile, which in turn controls the grid (finite difference mesh) for TEXSTAN

=0.0900    This value is recommended for most boundary layer flows. It is explained in considerable detail in the external flows: initial profiles sections of this website.

• tstag - this variable is the free stream temperature (static or stagnation, depending on kfluid and kgeom)

=432.0    This is the free stream stagnation temperature (degrees Kelvin) for the variable property (kfluid=2) Daniels and Browne turbine blade flow.

• vapp - this variable is the approach velocity, needed for extracting the initial value of the free stream turbulence kinetic energy from the tuapp variable for use with one- and two-equation turbulence models.

=146.40    This variable is not required for the mixing-length turbulence model. For extension of the problem statement to a one-equation or two-equation turbulence model, this is the correct approach velocity for the Daniels and Browne turbine blade. Note that it can not be left blank because it is part of the structured data input for TEXSTAN.

• tuapp - this variable is the free stream turbulence intensity and it used to establish the initial value of the free stream turbulence kinetic energy for one- and two-equation turbulence models at the location xstart

=3.0    This variable is not required for the mixing-length turbulence model. For extension of the problem statement to a one-equation or two-equation turbulence model, this is the estimated turbulence intensity for the Daniels and Browne turbine blade in the stagnation region. Note that it can not be left blank because it is part of the structured data input for TEXSTAN.

• epsapp - this variable establishes the initial value of the free stream turbulence dissipation for two-equation turbulence models at the location xstart

=0.0    This variable is not required for the mixing-length turbulence model. For extension of the problem statement to a two-equation turbulence model, this is the estimated turbulence dissipation. Because it is set =0.0, TEXSTAN will estimate the value (see modeling: two-equation model section of this website). From the Daniels and Browne paper a value of 8231.0 is recommended. Note that it can not be left blank because it is part of the structured data input for TEXSTAN.