Mechanical Engineering - The University of Texas at Austin

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The **initial conditions** are the initial profiles of velocity and all other dependent variables (for example temperature or stagnation enthalpy, mass fraction of A in an A-B binary mixture, turbulence kinetic energy, or turbulence dissipation), and they obey their respective transport equation boundary conditions. At a **surface** for both external flows and internal flows, for velocity will be zero, reflecting the no-slip condition; for the energy equation it will be either a specified surface temperature or surface heat flux (appropriately converted to enthalpy for variable properties); for mass transfer (selected geometries) it will be the surface mass concentration or mass fraction of A; for turbulence kinetic energy the value will be zero, reflecting both the no-slip condition and no penetration condition (the impermeable surface); for turbulence dissipation, the surface boundary condition is preprogrammed into TEXSTAN to be meet the unique requirements of its transport equation. At a **symmetry line**, for internal flows with geometries that have geometrical similarity (pipe flow and symmetrical thermal boundary condition planar duct flow), the zero-gradient condition is applied to all dependent variables. For the **free stream** and all external flows the free stream velocity at x = xstart is interpolated from the boundary condition tables; for the energy equation it is set by the tstag variable (and interpreted by TEXSTAN as free stream temperature for constant properties and stagnation temperature for variable properties); for mass transfer (selected geometries) it will be the interpolated free stream mass concentration or mass fraction of A; and for the turbulence variables, the free stream conditions are set by the input variables tuapp and epsapp.

The **solution domain** is where the boundary layer equations are solved, and it is bounded between two surfaces, called the I-surface and the E-surface. Integration of the transport equations begins at the user-specified x location called xstart where the initial conditions are specified, and integration is stopped at the user-specified x location called xend. The stop location is arbitrary, and it reflects the mathematical parabolic nature of the governing equations.

For external flows, the I-surface will always be the no-slip surface (wall) and the E-surface will always be the free stream. This E-surface is quasi-bounded, meaning it must be bounded for any integration step, but it can be moved as the integration proceeds to ensure a zero-gradient condition for each dependent variable. This is discussed in detail in the **entrainment** part of the external flows: integration control section of this website.

For internal flows, the flow domain is completely bounded in the cross-flow direction. For pipe flows and symmetrical thermal boundary condition planar ducts, the centerline is the I-surface and the no-slip surface is the E-surface. For asymmetrical thermal boundary condition planar ducts and for the annular geometry, the I- and E-surfaces are the inner and outer no-slip surfaces respectively.

The **initial profiles** are introduced into a TEXSTAN dataset in one of two ways:

- TEXSTAN auto-generates the initial profiles
- the user supplies the initial profiles

The **finite difference mesh** is created using the initial velocity profile, u(y), and the same distribution of y locations is used for the other dependent variables. Therefore, it is extremely important that the distribution of profile points in any input profile be correct, to help guarantee a grid-independent numerical solution. If the initial profiles are user-supplied (often from experimental data), TEXSTAN will interpolate these profiles to insure an appropriate distribution of initial profile points to satisfy the requirements of the numerics. The details of establishing the mesh is described in the overview: numerical accuracy section of the website.

TEXSTAN Auto-Generated Profiles - External Flows - The auto-generated profiles for **laminar flow** are the from the Falkner-Skan m=0 solution (Blasius profiles), which requires that the free stream velocity be constant and from the Falkner-Skan m=1 solution (2-D stagnation point profiles), which requires that the free stream velocity linearly accelerate. The stagnation-point profile can be applied as a first approximation to both the cylinder-in-crossflow and to the cylindrical leading-edge region of a turbine blade (airfoil). For **turbulent flow** the auto-generated profiles are from the classical power-law profiles with corrections for log-region behavior. For the energy equation, the temperature profile is first constructed, and then it is converted to a stagnation enthalpy profile if variable properties are being used. Specific details of how the various profiles are constructed are found in the external flows: initial profiles section of this website.

kstart | external flow initial profiles |
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=3 | turbulent velocity and temperature profiles for kgeom=1,2,3; includes k and ε profiles |

=4 | Blasius velocity and temperature profiles for kgeom=1 |

=5 | Falkner-Skan m=1 (2-D stagnation point) velocity and temperature profiles for kgeom=1 |

=6 | cylinder-in-crossflow velocity and temperature profiles for kgeom=1 |

=7 | turbine blade velocity and temperature profiles for kgeom=1; |

=9 | flat profiles of all variables for kgeom=1 (used on highly rough surfaces) |

TEXSTAN Auto-Generated Profiles - Internal Flows - The auto-generated profiles for **laminar flow** are designed to meet one of three flow cases: (a) the combined entry flow where the velocity and temperature profiles are flat; (b) the unheated starting length flow where the velocity profile is hydrodynamically fully-developed and the temperature profile is flat; and (c) the thermally fully-developed flow. The auto-generated profiles for **turbulent flow** are designed to meet one of two flow cases: combined entry flow (flat profiles of all variables) or thermally fully-developed flow where the profiles are approximated by the classic power-law profiles with corrections for log-region behavior. For the energy equation, the temperature profile is first constructed, and then it is converted to a stagnation enthalpy profile if variable properties are being used. Specific details of how the various profiles are constructed are found in the internal flows: initial profiles section of this website.

**Note:** for higher order turbulence models the entry flow profile choice is best. It is extremely difficult to establish fully-developed turbulence variable profiles because the profiles are turbulence-model specific, and by permitting the turbulent flow to establish itself within the combined entry region, there will be no influence of incorrect starting initial turbulence profiles on the friction and heat transfer solutions.

kstart | internal flow initial profiles |
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=1 | flat entry profiles of velocity and temperature for kgeom=4,5,6,7; note: includes k and ε profiles |

=2 | laminar hydrodynamically fully-developed velocity profile with either a flat or thermally fully-developed temperature profile for kgeom=4,5,6,7 |

=3 | turbulent velocity and temperature profiles for kgeom=4,5,6; includes k and ε profiles |

=11 | Couette flow velocity and temperature profiles for kgeom=6, laminar or turbulent flow |

User-Supplied Profiles for External Flows - This option is primarily designed to permit a more accurate numerical simulation of experimental convective heat and momentum data. There are two options to permit the user to input experimental initial profiles, and the specific details of how the various profiles are constructed are found in the external flows: initial profiles and internal flows: initial profiles sections of this website. The two options include:

kstart | special user-supplied initial profile options |
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=0 | primarily designed for restart of TEXSTAN based on a dump of profiles from a previous calculation |

=10 | primarily designed to input experimental profiles, selected kgeom and turbulence models |

When kstart=0 is selected, the user provides the input profile and TEXSTAN does nothing to correct the profile(s). This option is rarely used, except to re-read a set of profiles that TEXSTAN has generated and printed by way of the option k10=21-24 and the input variable bxx. For example, setting k10=24 permits a set of internal flow profiles to be written to file ftn73.txt at a location where bxx is defined as x/D_{h}. This file can then be used as-is as a user-supplied set of profiles to TEXSTAN as part of the input dataset (along with kstart=0). This option has proved useful for working with two-equation turbulence models.

When kstart=10, the user provides a set of u(y) data points (typically 15=20 from experimental data) and TEXSTAN interpolates the profile using the same algorithm for data point distribution that is used when TEXSTAN auto-generates a set of initial profiles. A temperature profile can also be interpolated and expanded. If the two-equation model of turbulence is being used, there is a special provision to auto-generate a dissipation profile using information from the experimental turbulence kinetic energy and velocity profiles.