## New integral and differential computational procedures for incompressible wall-bounded turbulent flows

##### Abstract

Three new computational procedures are presented for the simulation of
incompressible wall-bounded turbulent flows. First, an integral method based on the strip integral method has been developed for the solution of three-dimensional turbulent boundary layer flows. The integral equations written in a general form using non-orthogonal streamline coordinates include the turbulent shear stress at the upper limit of an inner strip inside the boundary-layer. The shear stress components are modeled using the Boussinesq assumption, and the eddy viscosity is defined explicitly as in differential methods. The turbulence modeling is not hidden in opaque empirical correlations as in conventional integral methods. A practical
four-parameter velocity profile has been established based on the Johnston Law of the Wall using a triangular model for the crosswise velocity. Two strips are used to solve for the four unknowns: skin friction coefficient, wall crossflow angle, boundary .. layer thickness, and location
of maximum crosswise velocity. The location of maximum crosswise velocity proves to be a natural and adequate parameter in the formulation, but it is numerically sensitive and has a strong influence on the wall crossflow angle. Good results were obtained when compared to
predictions of other integral or differential methods.
Secondly, two computational procedures solving the Reynolds Averaged Navier -Stokes equations for 20 and 3D flows respectively have also been developed using a new treatment of the near-wall region. The flow is solved down to the wall with a slip velocity based on Clauser's idea of a pseudolaminar velocity profile. The present idea is different from the wall function methods and does not require a multi-layer eddy viscosity model. The solution of the equations of motion is obtained by the Finite Element Method using the wall shear stress as a boundary condition along solid surfaces, and using the Clauser ou~er region model for the eddy viscosity. The wall shear stress distribution is updated by solving integral equations obtained from the enforcement of conservation of mass and momentum over an inner strip in the near wall region. The Navier-Stokes solution provides the necessary information to the inner strip
integral formulation in order to evaluate the skin friction coefficif~nt for 2D flows, or the skin friction coefficient and the wall crossflow angle for 3D flows. The procedures converge to the numerically -exact- solution in a few iterations depending on the accuracy of the initial guess for the wall shear stress. A small number of nodes is required in the boundary-layer to represent adequately the physics of the 1:low, which proves especially useful for 3D calculations. Excellent results were obtained for the 2D simulations with a simple eddy viscosity model. 3D calculations gave good results for the turbulent boundary-layer flows considered here.
The present methods were validated using well-known experiments chosen for the STANFORD conferences and EUROVISC workshop. The 2D numerical predictions are compared with the experimental measurements obtained by Wieghardt-Tillmann, SamuelJoubert, and Schubauer-Klebanoff. For the 3D analyses, the numerical predictions obtained by the strip-integral method and the Finite Element Navier-Stokes Intl3gral Equation procedure are validated IJsing the Van den Berg-Elsenaar and MOiler-Krause experiments.

##### Collections

- Doctoral Dissertations [13257]

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