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Projecte llegit

Títol: Numerical solution of the Bounadry Layer Equations

Estudiants que han llegit aquest projecte:

Director/a: MELLIBOVSKY ELSTEIN, FERNANDO PABLO

Departament: FIS

Títol: Numerical solution of the Bounadry Layer Equations

Data inici oferta: 08-02-2022     Data finalització oferta: 08-10-2022


Estudis d'assignació del projecte:
    GR ENG SIST AEROESP
Tipus: Individual
 
Lloc de realització: EETAC
 
Paraules clau:
boundary layer, incompressible, two-dimensional, numerical solver
 
Descripció del contingut i pla d'activitats:
The Navier-stokes equations may be parabolised by the application of the boundary layer equation. In this project, the resulting two-dimensional boundary layer equations will be expressed in the stream-function formulation and a code will be developed for their solution under arbitrary outer inviscid flow conditions.

The work plan is as follows:

0) Literature review on boundary layer equations and numerical methods for partial differential equations.

1) Derivation of the two-dimensional incompressible streamfunction formulation of the boundary layer equations, including the Falkner-Skan transformation.

2) Implement a central finite differences discretisation in the wall-normal coordinate and upstream final differences in the streamwise coordindate.

3) Implement a nonlinear solver (e.g. Newton method) to solve the resulting system of algebraic equations.

4) Consider a mapping for the semi-infinite domain and spectral methods (e.g. Chebychev collocation) for the wall-normal discretisation.

5) Consider implementing also time evolution.
 
Overview (resum en anglès):
The aim of this project is to develop a code that is capable of solving numerically the parabolised Navier-Stokes equations that govern the flow dynamics within two-dimensional boundary layers. Using a self-similarity scaling on the streamfunction formulation and given appropriate upstream and inviscid outer flow boundary conditions, the code solves the boundary layer and computes its characteristic properties.

To begin with, the two-dimensional boundary layer equations have been cast in the streamfunction formulation and a Falkner-Skan-type coordinate change has been applied to express them in similarity variables. Next, the resulting third order equation has been reduced to first order following a standard approach, and the system is discretized in space using finite differences.

The code has been tested against benchmark solutions for validation. The Blasius solution, which develops on a flat plate at zero incidence, and the stagnation point laminar boundary layer solution have been satisfactorily reproduced. Some problems previously solved with the approximate integral method have been revisited using the code to check the accuracy of the former.

The code has also been adapted to accept outer flow boundary conditions in the form of both closed-form mathematical expressions or discrete streamwise samplings of the inviscid outer streamwise velocity distribution. A simple turbulence model has also been coded to resolve turbulent as well as laminar boundary layers and a criterion for natural transition has also been implemented. Typical behavior of turbulent boundary layers, such as their tendency to resist separation better than laminar boundary layers, is duly predicted.

Finally, inviscid flow solutions past airfoils obtained with the software Xfoil have been fed into the boundary layer code to compute friction drag and detect separation. Results agree well with the literature, which further validates the accuracy of the boundary layer code

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