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AE 6345

Turbulence

Homework 1September 10, 2015

Due: September 17, 2015 in class

Let us examine certain well-established equations in fluid/gasdynamics but by allowing for disturbances andwhich can be tractable as homework exercises.

1. Low-speed wind tunnels have a bellmouth to accelerate the flow. This accelerate serves to dampen outturbulence and to improve the flow quality. Lets analyze this. Consider the Bernoulli equation which is usedfor incompressible flows

=+ 122

Let us apply Reynolds averaging with the usual assumption of stationarity, namely that a generic property

()=+ ()Suppose we now simplify the problem further by claiming that

()= as is commonly done.

a. Express in terms of the standard deviations of the other properties, noting that the flow isincompressible. It is preferable to normalize your result by the mean dynamic pressure

= 122

i.e.,

=1

where you have to fill in that.

b.

Consider a zero pressure gradient turbulent boundary layer. Surface pressure data from Speaker and

Ailman (NASA CR 486, 1966)* at Mach 0.42 shows that 8 103. Assuming that the static

pressure at the lower portion of the boundary layer shows similar fluctuations and assuming that

fluctuations of total pressure are negligible,obtain a numerical value for .

c. Comment on whether the pitot probe can be used for obtaining the mean velocity profile in anincompressible turbulent boundary layer.

d. Consider now that a turbulent flow with an incoming velocity of 50 m/s encounters an adverse pressuregradient so that the pressure drops from 100 kPa to 90 kPa over 1 m. Assume a constant density of 1.2

kg/m3. What is the velocity at the exit? If it is given that the incoming velocity fluctuations have astandard deviation of 1% of the mean incoming velocity, estimate the standard deviation of the velocity

fluctuations at the exit as a percentage of the mean exit velocity.

*Available upon request or you can do an online search.How is this justifiable?

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2. Consider velocity fluctuations across a shock wave. At high speeds, disturbances are associated with pressure,temperature and velocity, these being known as acoustic, entropy and vortical disturbances. The pressure ratioacross a shock wave can be written as

21 = 1 +

121 1

21

Ignore for simplicity entropic disturbances. Using the same Reynolds averaging procedure as before, derive a

(messy) expression for the vortical disturbances downstream of the shock wave in terms of the pressuredisturbances both upstream and downstream and the vortical disturbances upstream. You can either use primesto indicate disturbances or use the standard deviation for notational purposes. Note that this equation cannot be

solved for the downstream vortical disturbances 2 (or ) without further assumptions. One that is typicallyused is known as the strong Reynolds analogy (good luck).

3. In like manner, consider the Mach number of a perfect gas

=

For analysis, this can be rewritten as

2 =2

Morkovins hypothesis states that the turbulence field of a compressible flow can be treated to behave like that of

an incompressible flow as long as the turbulent Mach number 0.3. Perform a Reynolds averaging procedureto the Mach number definition and for values of 2 of 0.01 and 0.10 comment on the validity of thehypothesis. For simplicity, assume that the temperature fluctuations are negligible.

Bonus: What if the temperature fluctuations are not negligible? Temperature spottiness is especially notorious in

heated wind tunnels, e.g., NASA Langleys 8 Foot High Temperature Tunneland the NASA Glenns HypersonicTunnel Facility.

http://www.nasa.gov/centers/langley/news/factsheets/windtunnels.html http://www.grc.nasa.gov/WWW/Facilities/ext/htf/index.html