0.8 Multidimensional laminar flow (Page 8/8)

Transport phenomena Page 8 / 8

\begin{matrix} C_{f} & \equiv & \frac{τ_{x y}}{ρ {U_{\infty}}^{2}} = \frac{μ}{ρ {U_{\infty}}^{2}} {(\frac{\partial v_{x}}{\partial y}|}_{y = 0} \\ = & \frac{2}{\sqrt{8 π}} \frac{1}{\sqrt{U_{\infty} x / ν}} \\ = & \frac{0.3989}{\sqrt{{Re}_{x}}} \end{matrix}

This drag coefficient differs from the Blasius solution only by the coefficient of 0.332 in the exact solution.

The evolution of the boundary layer velocity profile with equal increments of distance from the leading edge of the plate is illustrated in the following figure. It is suggested that the student execute the plate.m file in the boundary directory to view the movie of the evolution of the velocity profile and to examine the equations used in the calculations.

Blasius solution for boundary layer flow past a flat plate

The previous solutions were instructive in that they illustrated the correspondence with the diffusion of motion from a plate to the bulk fluid. However, the approximate solutions did not exactly satisfy either the continuity equation or the equations of motion. Blasius used the stream function to exactly satisfy continuity equation. The equations of motions were simplified by the boundary layer assumption that the thickness of the boundary layer is small compared to the distance from the leading edge of the plate. Also, the pressure gradient is zero for this case.

\begin{matrix} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \\ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = ν \frac{\partial^{2} u}{\partial y^{2}} \\ y = 0 : u = v = 0 \\ y \to \infty : u = U_{\infty} \end{matrix}

Assume that the dimensionless velocity profile can be expressed as a function of a similarity variable.

\frac{u}{U_{\infty}} = u^{*} = u^{*} (\frac{y}{δ (x)})

The approximate solution had the following form:

\frac{v_{x}^{(1)}}{U_{\infty}} = \erf (\frac{y}{\sqrt{8 ν x / U_{\infty}}})

This suggests a similarity variable:

\begin{matrix} η = \frac{y}{\sqrt{ν x / U_{\infty}}} = y \sqrt{\frac{U_{\infty}}{ν x}} \\ \frac{\partial η}{\partial y} = \sqrt{\frac{U_{\infty}}{ν x}}, \frac{\partial η}{\partial x} = \frac{- η}{2 x} \end{matrix}

The equation of motion is expressed in terms of the stream function.

\begin{matrix} u = \frac{\partial ψ}{\partial y}, v = \frac{- \partial ψ}{\partial x} \\ \frac{\partial ψ}{\partial y} \frac{\partial^{2} ψ}{\partial x \partial y} - \frac{\partial ψ}{\partial x} \frac{\partial^{2} ψ}{\partial y^{2}} = ν \frac{\partial^{3} ψ}{\partial y^{3}} \end{matrix}

Assume that the stream function is a function only of the similarity variable.

\begin{matrix} ψ = ψ (η) \\ u = \frac{\partial ψ}{\partial y} = \frac{d ψ}{d η} \frac{\partial η}{\partial y} = \sqrt{\frac{U_{\infty}}{ν x}} \frac{d ψ}{d η} \end{matrix}

Make dimensionless:

\begin{matrix} U_{\infty} u^{*} = \sqrt{\frac{U_{\infty}}{ν x}} ψ_{o} \frac{d ψ^{*}}{d η} \\ u^{*} = \frac{ψ_{o}}{\sqrt{ν U_{\infty} x}} \frac{d ψ^{*}}{d η} \\ \Rightarrow ψ_{o} = \sqrt{ν U_{\infty} x} \end{matrix}

The dimensionless stream function is expressed as a function of only the similarity variable.

\begin{matrix} ψ^{*} & = & f (η) \\ ψ & = & \sqrt{ν U_{\infty} x} f (η) \\ u & = & U_{\infty} \frac{d f}{d η} = U_{\infty} f^{'} \\ v & = & \frac{- \partial ψ}{\partial x} \\ = & - \frac{\partial}{\partial x} [\sqrt{ν U_{\infty} x} f (η)] \\ = & \frac{1}{2} \sqrt{\frac{ν U_{\infty}}{x}} [η f^{'} - f] \\ \frac{\partial u}{\partial x} & = & - \frac{U_{\infty}}{2 x} η f^{''} \\ \frac{\partial u}{\partial y} & = & U_{\infty} \sqrt{\frac{U_{\infty}}{ν x}} f^{''} \\ \frac{\partial^{2} u}{\partial y^{2}} & = & \frac{U_{\infty}^{2}}{ν x} f^{'''} \end{matrix}

Substituting the above equations into the equation of motion and cancellation of two terms results in the following ordinary differential equation.

\begin{matrix} 2 f^{'''} + f f^{''} = 0 \\ f (0) = f^{'} (0) = 0 \\ f^{'} (η \to \infty) = 1 \end{matrix}

This is a third order ODE with two conditions at $η \to \infty$ and one condition at . It is convenient to solve it as a set of first order ODEs with initial conditions, two of which are specified and the third adjusted such as to satisfy the condition at infinity.

Y = [\begin{matrix} f \\ f^{'} \\ f^{''} \end{matrix}], \frac{d Y}{d η} = [\begin{matrix} f^{'} \\ f^{''} \\ f^{'''} \end{matrix}] = [\begin{matrix} f^{'} \\ f^{''} \\ - f f^{''} / 2 \end{matrix}] = [\begin{matrix} Y_{2} \\ Y_{3} \\ - Y_{1} Y_{3} / 2 \end{matrix}]

This set of ODEs can be solved numerically by one of the ODE solvers and the initial value of $f^{''}$ iterated until the boundary condition at infinity is matched. The code of this calculation is in the boundary directory as files, blasius.m , blasiusf.m , and balsiusd.dat .

Assignment 9.2: boundary layer flow past a wedge

Derive the equations for boundary layer flow past a wedge. Use a factor of in the denominator of the similarity variable to be in keeping with contemporary textbooks.
Use the code in the boundary directory of the CENG 501 website to solve the Flakner-Skan equation.
Plot the velocity profiles as a function of the similarity variable for different angles of the wedge relative to the approaching free-stream velocity. Replace the parameter beta with the angle in degrees.
Illustrate the boundary layer thickness by plotting contour lines of 10
Plot the equally spaced streamlines for the same cases.

Assignment 9.3: flow in a wedge with zero shear at $θ = 0 .$

Start from the continuity and Navier-Stokes equation and derive the equations for the flow field near a corner for flow in a wedge of fluid with no slip on one side and zero shear stress along

θ = 0

. List all your assumptions.

Derive expressions for the stream function, velocity, and pressure.
For what distance from the corner is the solution valid?
What normal stress is required to keep the = 0 surface flat?
If the surface of zero shear stress can sustain only finite normal stress, in which way will the surface deform? Recognize that the no-slip surface can travel in either direction.
After deriving the equations, view and plot the flow field for various angles using wedge.m file in the creeping directory of the CENG 501 website.

Assignment 9.4: rise of a spherical, inviscid bubble in a liquid.

Start from the continuity and Navier-Stokes equation and derive the equations for the flow field of a spherical, inviscid bubble rising in a liquid by buoyancy. List all assumptions.

Derive expressions for the stream function, velocity, and pressure.
Derive the expression for the terminal rise velocity. How does it differ from the case with no slip?

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Source: OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1

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