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364 On the Calculation of the Velocity and Temperature Distributions for Flow along a Flat Plate By L. Howarth, B.A., B.Sc., Gonville and Caius College, Cambridge (Communicated by G. I. Taylor, F.R.S.—Received November 5, 1935) Introduction Karman and Prandtl were the first investigators to publish theoretical results for problems of turbulent flow involving plane boundaries. Before considering any particular problem the general considerations of these writers will be outlined.
Flow Along a Flat Plate 365 (ii) that / is proportional to (du/dy) /(d u/dy2), i.e., 2 I = K (du I dy) l (d2u /dy2) (2) where K is a constant, (iii) that the shearing stress t is equal to p/2 (du/dy)2 a constant factor being supposed absorbed in K in (2).
366 L. Howarth / does vanish at the wall and so we put Kyf as the first term in the series for /. Equation (3) may be taken therefore as a first approximation to the flow in the neighbourhood of a plane wall in the presence of any pressure gradient parallel to the wall.
Flow Along a Flat Plate 369 of motion as the previous one. We shall refer to this as the equation of motion on the two-dimensional vorticity theory. This equation of motion leads to a form for the temperature distribution different from that of the velocity distribution. It has, therefore, usually been held that the vorticity theory is of little value for Wall Turbulence. It is interesting to determine whether, in fact, the temperature and velocity distributions found from the two-dimensional vorticity theory are very different. This is the object of this paper.
370 L. Howarth Details Velocity Distribution—Denote by x distance measured parallel to the plate from the leading edge, f by ydistance measured normal to the plate, by u and v the velocities in the directions x and y increasing, and by U the velocity of flow in the main stream. The equations of motion on the momentum and vorticity theories are 21 u TT- + V -5- — (7) ox oy - ( % u du . du 12 du d2u u te + v lTy = l (8) respectively.
Flow Along a Flat Plate 371 The assumption of similarity implies that the equation of motion reduces to an ordinary differential equation for F with 73 as independent variable. With Prandtl’s assumption that is a constant multiple of y or Kar man’s form, this assumption gives <f> (x) = x. The form for u obtained from this value of <f> yields a constant value for the skin friction and obviously a boundary layer thickness increasing linearly with the distance from the leading edge. Neither of these results is in agreement with experiment and, presumably, the correct conclusion is that there is no similarity of the velocity profiles.
372 L. Howarth respectively where dashes denote differentiations with regard to 73. On putting S = (15) these reduce to F" (F + 2CF" + 2X? F'") = 0, (l6) F" (F + F'") = 0, (17) where dashes now denote differentiations with regard to C We neglect the solution F" = 0, which corresponds to the velocity distribution u = constant. Each equation has therefore three inde pendent solutions, the general solution being given by a linear com bination of multiples of them.
Flow Along a Flat Plate 373 if a and the value of v\ for which = U and du/dy = 0, were known from experiment, then A and B would be entirely determined. Thus it appears that the two constants a and y)0 together with the function <f> (x) are not determined by this investigation.