Next: Steady State Laminar Boundary
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In developing a mathematical theory of boundary layers, the first step is to show the existence, as the Reynolds number R tends to infinity, or the kinematic viscosity
tends to zero, of a limiting form of the equations of motion, different from that obtained by putting
in the first place. A solution of these limiting equations may then reasonably be expected to describe approximately the flow in a laminar boundary layer for which R is large but not infinite. This is the basis of the classical theory of laminar boundary layers.
The full equation of motion for for a two-dimensional flow are:
![\begin{displaymath}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+...
...}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right),
\end{displaymath}](img8.gif) |
(1) |
![\begin{displaymath}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+...
...}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}\right),
\end{displaymath}](img9.gif) |
(2) |
![\begin{displaymath}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} = 0,
\end{displaymath}](img10.gif) |
(3) |
where the x and y variables are, respectively, the horizontal and vertical coordinates, u and v are, respectively, the horizontal and vertical fluid velocities and p is the fluid pressure. A wall is located in the plane y = 0. We consider non-dimensional variables
![\begin{displaymath}x' = \frac{x}{L},
\end{displaymath}](img11.gif) |
(4) |
![\begin{displaymath}y' = \frac{y}{\delta},
\end{displaymath}](img12.gif) |
(5) |
![\begin{displaymath}u' = \frac{u}{U},
\end{displaymath}](img13.gif) |
(6) |
![\begin{displaymath}v' = \frac{v}{U}\frac{L}{\delta},
\end{displaymath}](img14.gif) |
(7) |
![\begin{displaymath}p' = \frac{p}{\rho U^{2}},
\end{displaymath}](img15.gif) |
(8) |
![\begin{displaymath}t' = t\frac{U}{L},
\end{displaymath}](img16.gif) |
(9) |
where L is the horizontal length scale,
is the boundary layer thickness at x = L, which is unknown. We will obtain an estimate for it in terms of the Reynolds number R. U is the flow velocity, which is aligned in the x-direction parallel to the solid boundary. The non-dimensional form of the governing equations is:
![\begin{displaymath}\frac{\partial u'}{\partial t'}+u'\frac{\partial u'}{\partial...
...ac{L^{2}}{\delta^{2}}\frac{\partial^{2}u'}{\partial (y')^{2}},
\end{displaymath}](img18.gif) |
(10) |
![\begin{displaymath}\frac{\partial v'}{\partial t'}+u'\frac{\partial v'}{\partial...
...L}{\delta}\right)^{2}\frac{\partial^{2}v'}{\partial (y')^{2}},
\end{displaymath}](img19.gif) |
(11) |
![\begin{displaymath}\frac{\partial u'}{\partial x'}+\frac{\partial v'}{\partial y'} = 0,
\end{displaymath}](img20.gif) |
(12) |
where the Reynolds number for this problem is
![\begin{displaymath}R = \frac{UL}{\nu}.
\end{displaymath}](img21.gif) |
(13) |
Inside the boundary layer, viscous forces balance inertia and pressure gradient forces. In other words, inertia and viscous forces are of the same order, so
![\begin{displaymath}\frac{\nu}{UL}\left(\frac{L}{\delta}\right)^{2} = O(1) \Rightarrow \delta = O(R^{-1/2}L).
\end{displaymath}](img22.gif) |
(14) |
Now we drop the primes from the non-dimensional governing equations and with equation (2.14) we have
![\begin{displaymath}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+...
...al^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}},
\end{displaymath}](img23.gif) |
(15) |
![\begin{displaymath}\frac{1}{R}\left(\frac{\partial v}{\partial t}+u\frac{\partia...
...}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}\right),
\end{displaymath}](img24.gif) |
(16) |
![\begin{displaymath}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} = 0.
\end{displaymath}](img25.gif) |
(17) |
In the limit
,
the equations above reduce to:
![\begin{displaymath}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+...
...{\partial p}{\partial x}+\frac{\partial^{2}u}{\partial y^{2}},
\end{displaymath}](img27.gif) |
(18) |
![\begin{displaymath}-\frac{\partial p}{\partial y} = 0,
\end{displaymath}](img28.gif) |
(19) |
![\begin{displaymath}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} = 0.
\end{displaymath}](img25.gif) |
(20) |
Notice that according to equation (2.19), the pressure is constant across the boundary layer. In terms of dimensional variables, the system of equations above assume the form:
![\begin{displaymath}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+...
...artial p}{\partial x}+\nu\frac{\partial^{2}u}{\partial y^{2}},
\end{displaymath}](img29.gif) |
(21) |
![\begin{displaymath}-\frac{1}{\rho}\frac{\partial p}{\partial y} = 0,
\end{displaymath}](img30.gif) |
(22) |
![\begin{displaymath}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} = 0.
\end{displaymath}](img25.gif) |
(23) |
To solve the system of equations above we need to specify initial and boundary conditions.
Next: Steady State Laminar Boundary
Up: No Title
Previous: Introduction.
Karl P Burr
2003-03-12