CSU MECH 344 - HW6 - Solutions (5 pages)

PROBLEM 6 2 KNOWN Form of the velocity and temperature profiles for flow over a surface FIND Expressions for the friction and convection coefficients SCHEMATIC ANALYSIS The shear stress at the wall is s u y y 0 A 2By 3Cy 2 y 0 A Hence the friction coefficient has the form Cf Cf s 2 2 u 2A 2 u 2A 2 u The convection coefficient is 2 k f T y y 0 k f E 2Fy 3Gy y 0 h Ts T D T h k f E D T COMMENTS It is a simple matter to obtain the important surface parameters from knowledge of the corresponding boundary layer profiles However it is rarely a simple matter to determine the form of the profile PROBLEM 6 5 KNOWN Variation of hx with x for laminar flow over a flat plate FIND Ratio of average coefficient h x to local coefficient hx at x SCHEMATIC ANALYSIS The average value of hx between 0 and x is 1 x C x h x dx x 1 2dx x 0 x 0 C 1 2 2x 2Cx 1 2 x 2h x hx hx hx Hence hx 2 hx COMMENTS Both the local and average coefficients decrease with increasing distance x from the leading edge as shown in the sketch below PROBLEM 6 19 KNOWN Air speed and temperature in a wind tunnel 8 FIND a Minimum plate length to achieve a Reynolds number of 10 b Distance from leading edge at which transition would occur SCHEMATIC ASSUMPTIONS 1 Isothermal conditions Ts T 6 2 PROPERTIES Table A 4 Air 25 C 298K 15 71 10 m s ANALYSIS a The Reynolds number is Re x u x u x 8 To achieve a Reynolds number of 1 10 the minimum plate length is then 8 6 2 Re x 1 10 15 71 10 m s L min u 50 m s L min 31 4 m b For a transition Reynolds number of 5 10 xc Re x c u 5 5 105 15 71 10 6 m 2 s x c 0 157 m 50 m s COMMENTS Note that x c Re x c L ReL This expression may be used to quickly establish the location of transition from knowledge of Re x c and Re L