7.3. One way to improve the mixing-length model, that is, to achieve a smoother overlap between measurements and the empirically adjusted curves suggested by the mixing-length model (Fig. 7.4), is to do away with the assumption that the eddy diffusivity epsi lon_(M) is zero in a layer of finite thickness y_(VSL). Instead, as proposed by van Driest [16], assume that epsi lon_(M) decays rapidly as y decreases and becomes zero strictly at the wall. Starting with the new mixing-length model,
l=kappa y(1-e^(-(y^(+))/(A^(+))))
and assuming that it is valid throughout the inner region defined by the constant shear stress postulate (7.33), develop the analytical means for calculating u^(+)as a function of y^(+),kappa , and A^(+). Setting kappa =0.4, show that the new u^(+)calculation fits the data of Fig. 7.4 smoothly if the new empirical constant A^(+)is approximately 25 .
7.3.One way to improve the mixing-length model, that is, to achieve a smoother overlap between measurements and the empirically adjusted
with the assumption that the eddy diffusivity em is zero in a layer of finite thickness yvst.Instead,as proposed by van Driest [16], assume that y decays rapidly as y decreases and becomes zero strictly at the wall. Starting with the new mixing-length model,
1=Ky1-ey*/A
and assuming that it is valid throughout the inner region defined by the constant shear stress postulate 7.33, develop the analytical means for calculating u*as a function of y+,K,and A*.Setting K=0.4,show that the new u* calculation fits the data of Fig. 7.4 smoothly if the new empirical constant A+is approximately 25.
