Assuming negligible convective and diffusive terms, the Reynolds stress transport equation simplifies to: Pij + Qij - Eij = 0
P = Epsilon
Considering the following modeling:
Qij = Qij + Yij + Wij
Phiij = 3c1(Epsilon)/(k)((2)/(3)k deltaij - tauij)
Psiij = alpha((2)/(3)P deltaij - Pij) + beta((2)/(3)P deltaij - Deltaij) - gamma(k)/(bar D)ij
Wij = (alpha0^((3)/(2)))/(kappa)[c1(Epsilon)/(k)(tauij - (2)/(3)k deltaij) + c2(Pij - Deltaij)](k^((3)/(2)))/(Epsilon y)
alpha = (6.666c2 + 8)/11
beta = (53.333c2 - 2)/11
gamma = 4(100c2 - 1)/55
c1 = 0.5 and c2 = 0.06
Show that:
(a) For a plane homogeneous shear layer, the stresses are: tau11 = 0.967k, tau22 = 0.487k, tau33 = 0.5467k, and tau12 = -0.33k
Hint: For this case, the only gradient is (del(u)1)/(delx2) = C, where C is constant. Also, neglect Wij.
(b) For a near-wall turbulent shear layer, the stresses are: tau11 = 1.177k, tau22 = 0.2467k, tau33 = 0.5767k, and tau12 = -0.24k
Hint: For this case, the only gradient is (del(u)1)/(delx2) = C, where C is constant. Do not neglect Wij.
(c) What is the ratio of normal stress anisotropy for plane channel flow?
