mullzer
Jun3-11, 03:17 AM
I am studying fluid dynamics by using adimensional analysis to obtain an expression for a dynamic (thermal) boundary layer in a poiseuille flow. The fluid nearest the boundary (y = 0) is said to have adimenional temperature T = 1 and as y tends to infinity, the temperature is T = 0.
The final adimensionalised problem usually take the form of the diffusion equation, e.g.
2y(dT/dx) = (d2T/dy2)
The method prefered by my superior is use "invariance groups"- i.e. to brake each variable in the problem into two parts: e.g. y = y*.y^ which leads to:
2(y*.y^) (T*/x*)(dT^/dx^) = (T*/y*2)(d2T/dy2)
The priniciple here is that in order for the original expression to hold true, all the * variables must cancel out to 1, leaving the original equation.
Equating the * terms brings the relation: y*= x*1/3.
The variable of similitude is then introduced, n= y/x(1/3), and the temperature can be written T(x,y) = f(n).
What i cant grasp is how to formulate this f(n). In this example, it takes the form:
f'' + (2/3)n2f' = 0.
I am not only interested in how this specific example was formulated but also the method in general as I am quite unfamiliar with it.
Thanks
The final adimensionalised problem usually take the form of the diffusion equation, e.g.
2y(dT/dx) = (d2T/dy2)
The method prefered by my superior is use "invariance groups"- i.e. to brake each variable in the problem into two parts: e.g. y = y*.y^ which leads to:
2(y*.y^) (T*/x*)(dT^/dx^) = (T*/y*2)(d2T/dy2)
The priniciple here is that in order for the original expression to hold true, all the * variables must cancel out to 1, leaving the original equation.
Equating the * terms brings the relation: y*= x*1/3.
The variable of similitude is then introduced, n= y/x(1/3), and the temperature can be written T(x,y) = f(n).
What i cant grasp is how to formulate this f(n). In this example, it takes the form:
f'' + (2/3)n2f' = 0.
I am not only interested in how this specific example was formulated but also the method in general as I am quite unfamiliar with it.
Thanks