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Chapter 6: Problem 99

Consider a fluid flowing over a flat plate at a constant free stream velocity. The critical Reynolds number is \(5 \times 10^{5}\) and the distance from the leading edge at which the transition from laminar to turbulent flow occurs is \(x_{\mathrm{cr}}=7 \mathrm{ft}\). Determine the characteristic length \(\left(L_{c}\right)\) at which the Reynolds number is \(1 \times 10^{5}\).

Short Answer

Expert verified
Question: Determine the characteristic length at which the Reynolds number is \(1 \times 10^5\) for a fluid flowing over a flat plate with the given critical Reynolds number (\(Re_{cr} = 5 \times 10^5\)) and the distance from the leading edge at which the transition from laminar to turbulent flow occurs (\(x_{cr} = 7 ft\)). Answer: The characteristic length at which the Reynolds number is \(1 \times 10^5\) is \(\frac{7}{5} ft\).

Step by step solution

01

Recall the Reynolds Number formula

The Reynolds number for flow over a flat plate can be expressed as: \[Re = \frac{\rho V L}{\mu}\] where \(Re\) - Reynolds number, \(\rho\) - fluid density, \(V\) - free stream velocity, \(L\) - characteristic length of the flow, and \(\mu\) - dynamic viscosity of the fluid.
02

Determine the free stream velocity in terms of the characteristic length

First, we need to find the free stream velocity in terms of the characteristic length \(L_c\). We are given the critical Reynolds number, which occurs at the distance \(x_{cr}\) from the leading edge. Knowing that \(Re_{cr} = 5 \times 10^5\), and \(x_{cr} = 7 ft\), we rearrange the Reynolds number formula to find the velocity as a function of the characteristic length: \[V = \frac{Re_{cr} \mu}{\rho x_{cr}}\] We do not have the values for \(\rho\) and \(\mu\), but since \(Re_{cr}\) already contains these values, we do not need them. We can simply express the unknown free stream velocity \(V\) as a function of characteristic length using this equation.
03

Determine the characteristic length at the given Reynolds number

Now, we are asked to find the characteristic length where the Reynolds number is \(1 \times 10^5\). We can write the Reynolds number formula for the given Reynolds number value and the characteristic length \(L_c\): \[1 \times 10^5 = \frac{\rho V L_c}{\mu}\] Rearranging the equation, we get: \[L_c = \frac{1 \times 10^5 \mu}{\rho V}\] Substitute the expression for \(V\) from step 2 into this equation: \[L_c = \frac{1 \times 10^5 \mu}{\rho \frac{Re_{cr} \mu}{\rho x_{cr}}}\] Simplify the equation: \[L_c = \frac{1 \times 10^5 x_{cr}}{Re_{cr}}\] Now plug in the given values \(x_{cr} = 7 ft\) and \(Re_{cr} = 5 \times 10^5\): \[L_c = \frac{1 \times 10^5 \times 7}{5 \times 10^5}\]
04

Calculate the characteristic length

Finally, we can calculate the characteristic length \(L_c\): \[L_c = \frac{7 \times 10^5}{5 \times 10^5} = \frac{7}{5} ft\] Thus, the characteristic length \(L_c\) at which the Reynolds number is \(1 \times 10^5\) is \(\frac{7}{5} ft\).

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Most popular questions from this chapter

Consider fluid flow over a surface with a velocity profile given as $$ u(y)=100\left(y+2 y^{2}-0.5 y^{3}\right) \mathrm{m} / \mathrm{s} $$ Determine the shear stress at the wall surface, if the fluid is \((a)\) air at \(1 \mathrm{~atm}\) and \((b)\) liquid water, both at \(20^{\circ} \mathrm{C}\). Also calculate the wall shear stress ratio for the two fluids and interpret the result.

Friction coefficient of air flowing over a flat plate is given as \(C_{f}=0.664(V x / \nu)^{-0.5}\), where \(x\) is the location along the plate. Using EES (or other) software, determine the effect of the air velocity \((V)\) on the wall shear stress \(\left(\tau_{w}\right)\) at the plate locations of \(x=0.5 \mathrm{~m}\) and \(1 \mathrm{~m}\). By varying the air velocity from \(0.5\) to \(6 \mathrm{~m} / \mathrm{s}\) with increments of \(0.5 \mathrm{~m} / \mathrm{s}\), plot the wall shear stress as a function of air velocity at \(x=0.5 \mathrm{~m}\) and \(1 \mathrm{~m}\). Evaluate the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

What is a similarity variable, and what is it used for? For what kinds of functions can we expect a similarity solution for a set of partial differential equations to exist?

What fluid property is responsible for the development of the velocity boundary layer? For what kind of fluids will there be no velocity boundary layer on a flat plate?

Will a thermal boundary layer develop in flow over a surface even if both the fluid and the surface are at the same temperature?
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