boundary conditions

TEXSTAN can handle a variety of different boundary conditions. For the energy equation the flow-direction variation of either the wall heat flux or the wall temperature can be specified. The mass flux at a wall can also vary in the flow direction for certain geometries. For geometries with two different walls, such as annuli or planar ducts, asymmetric thermal boundary conditions can be accommodated. For external flows, the free-stream velocity, rather than the pressure, is treated as a variable boundary condition, while the free stream stagnation enthalpy (variable properties) or free stream temperature (constant properties) is held constant. Mass transfer boundary conditions are quite specific and described in external flows: mass transfer.

Momentum Boundary Conditions - There are several classic types of momentum boundary conditions programmed into TEXSTAN. The no-slip (zero flow-direction velocity) boundary condition is required at all surfaces for both external flows and internal flows. The surface-normal velocity relates to transpiration and it is either set zero (solid or impermeable surface) or it is non-zero for the case of blowing or suction. Surface transpiration is permitted only for certain external flow geometries, and it is not permitted for internal flows. For internal flows in pipes (and between parallel planes with symmetrical thermal boundary conditions at each surface) the centerline boundary condition is a zero-velocity gradient condition.

All internal flow boundary conditions for momentum are pre-programmed into TEXSTAN.

Note: Even though various boundary conditions may be pre-programmed, they must be entered into the dataset boundary condition table as a "0.0" value, rather than being left blank, because TEXSTAN uses a structured data set style.

For external flow, the no-slip boundary condition is pre-programmed into TEXSTAN. The user can supply a surface-normal mass flux (see transpiration) for certain flow geometries. For the free stream velocity, the boundary condition is input as an array of values, obtained either from experimental data, an analytical distribution, or an Euler-generated (potential flow) solution. TEXSTAN then interpolates these values to provide a continuous distribution of free stream velocity for the boundary condition. To know the minimum number of data points required for the array, it is important to understand how TEXSTAN handles this array of values. For two points, TEXSTAN linearly interpolates, but for multiple points, it uses a spline-fit routine, which can lead to huge problems if the data are not differentially smooth (see overview: bc smoothness to understand this problem and the steps that are needed to ensure the data are smooth). 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.

Thermal Boundary Conditions - TEXSTAN has two types of thermal boundary conditions that affect the wall temperature and promote or suppress heat transfer. The first type controls the wall temperature (the Dirichlet type of boundary condition) and the second type controls the wall heat flux (the Neumann type of boundary condition). In either case, TEXSTAN permits the wall temperature or wall heat flux to vary in the flow direction. There is no difference between internal and external flow

A constant wall temperature boundary condition (isothermal) typically models a wall that either has some sort of phase change process occurring on the other side of the wall or the wall has a thermal mass (mass-specific heat product) that is large enough that it can be considered isothermal while exchanging heat with the fluid. A constant heat flux boundary condition typically models a wall that either has some sort of thermal radiation loading onto the wall or there is a heat flux source on the other side of the wall due to Joule heating or a nuclear source.

The surface thermal boundary condition for each surface (recall flow between thermally asymmetrical parallel plates and flow in an annulus have two surfaces) is input as an array of values. For each array, TEXSTAN then linearly interpolates adjacent sets of values, which leads to a piecewise-continuous distribution in the boundary condition, but no continuous first derivative. For example, to simulate a step-change in surface temperature or surface heat flux it is only necessary to provide two boundary condition values that are "epsilon" apart in the flow direction to create the step. A surface that is half adiabatic and half at some finite (non-zero) heat flux would require four boundary condition values - a first x-location value corresponding to the starting location of the surface (within the first half of the surface) that has a zero heat flux value, a second x-location value halfway along the surface, also with a zero heat flux value, a third x-location value that is "epsilon" further along the surface with the non-zero heat flux value, and finally a fourth x-location value at some defined surface end location. Because of the linear interpolation between adjacent array values, the surface heat flux will be a constant value (equal to zero) over the first half, some finite value but only over the "epsilon" flow distance, and then a constant value (equal to the nonzero heat flux value) over the last half of the surface.

Note: TEXSTAN can not mix the type of thermal boundary condition along the flow direction. It must be continuously either a surface temperature or a surface heat flux. There are some workarounds. For example, in external flows, the first part of a flow surface will behave adiabatic if its surface temperature is set equal to the same value as the free stream temperature (constant properties). For internal flows, the first part of the flow surface will behave adiabatic if its surface is set equal to the fluid inlet temperature.

Example - Hopefully this set of boundary conditions for the Blasius boundary layer will show how easily the boundary conditions are implemented in TEXSTAN. The geometry is a planar flat plate that is 0.2 m long, corresponding to an x-Reynolds number ~ 2 x 10⁵. The free stream velocity is u_∞ = 15 m/s , the free stream temperature is T_∞ = 300K , and the surface temperature is held constant at T_s = 295K . There is no surface transpiration, so the am arrays are set =0.0 We can see that the input variable array x(m) provides the beginning location and ending location that describe the plate's dimensions, fj(I,1,m) provides the surface temperature, and ubE(m) provides the free stream velocity at these two x(m) locations. Because the free stream temperature can never change for a constant-property calculation (or free stream total temperature for a variable property calculation) this value is input using the scalar variable (tstag) rather than within the boundary condition tables. The wall transverse radius array rw(m) is set to unity to permit it to divide out of the equations (recall the transport equations are in an axisymmetric formulation and this reduces the equations to Cartesian).

Note: (again) that even though various boundary conditions may be pre-programmed, they must be entered into the dataset boundary condition table as a "0.0" value, rather than being left blank, because TEXSTAN uses a structured data set style.

x(m)	rw(m)	aux1(m)	aux2(m)	aux3(m)
0.0	1.0	0.0	0.0	0.0
0.20	1.0	0.0	0.0	0.0

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	0.0	295.0
15.00	0.0	0.0
0.00	0.0	295.0
15.00	0.0	0.0