2. Dimensional analysis (for $K_{\infty }\neq 0$ )

The maximum velocity $u_{m}$ of the turbulent wall jet with the asymptotic kinematic momentum flux $K_{\infty }~(\text{m}^{3}~\text{s}^{-2})$ has the form

(2.1) $$\begin{eqnarray}u_{m}=f(x-x_{0},{\it\nu},K_{\infty }),\end{eqnarray}$$

where $x_{0}$ is the coordinate of the virtual origin and ${\it\nu}$ the kinematic viscosity. The asymptotic formula (2.1) is by definition independent of the slot width $b$ and the inlet momentum flux $K_{j}$ .

Dimensional analysis leads to

(2.2) $$\begin{eqnarray}\frac{u_{m}\;{\it\nu}}{K_{\infty }}=F(\mathit{Re}_{x}),\end{eqnarray}$$

where the Reynolds number is defined as

(2.3) $$\begin{eqnarray}\mathit{Re}_{x}=\frac{\sqrt{(x-x_{0})\;K_{\infty }}}{{\it\nu}}.\end{eqnarray}$$

The one and only curve in the representation of (2.2) is universal and valid for all possible plane turbulent wall jets.

Also the following dimensionless combinations can depend only on the Reynolds number:

(2.4) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}y_{0.5}/(x-x_{0}),\quad y_{m}/y_{0.5}\;(\text{figure 2}),\quad u_{m}\;y_{0.5}/{\it\nu}\;(\text{figure 5}),\quad \sqrt{(x-x_{0})K}/{\it\nu}\\ \mathit{Re}_{m}=u_{m}\;y_{m}/{\it\nu},\quad c_{f}/2={\it\tau}_{w}/{\it\rho}\;u_{m}^{2}\;(\text{figure 2}),\quad {\it\tau}_{w}\;{\it\nu}^{2}/{\it\rho}\;K_{\infty }^{2},\quad K/K_{\infty }\;(\text{figure 5}),\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $K$ is the (kinematic) momentum flux (see definition in (4.14)), ${\it\rho}$ the density, ${\it\tau}_{w}$ the wall shear stress, $y_{m}$ the wall distance of the maximum velocity and $y_{0.5}$ the wall distance of the point with velocity $u_{m}/2$ . All dimensionless combinations in (2.4) are also universal functions of $\mathit{Re}_{m}$ (examples in figure 2).

Figure 2.

Skin friction law $c_{f}(\mathit{Re}_{m})$ and thickness ratio $y_{m}/y_{0.5}$ for turbulent plane wall jets. —— after (4.28), - - - - after Hammond (Reference Hammond1982), – – – after Launder & Rodi (Reference Launder and Rodi1981), - $\cdot$ - $\cdot$ - asymptote,  $G=1$ in (4.28),  $\cdots \cdots$  after (6.1). Bars refer to experiments mentioned in table 1 presented in § 9; triangles are selected points for agreement with theory.

3. Flow model

The flow field consists of two parts: (i) the lower part $(y<y_{m})$ behaves like a turbulent boundary layer; (ii) the upper part $(y>y_{m})$ behaves like a half-free jet. Similar to turbulent boundary layers the lower part of the wall jet can be divided into three layers: the viscous wall layer, the overlap layer and the so-called defect layer. Hence, the model of the wall-jet flow field under consideration has a four-layer structure. Furthermore it is assumed that the velocity distribution is self-similar, but separately in each of the four layers.

In many experimental investigations of turbulent wall jets the results focus on the global functions $y_{m}(x),y_{0.5}(x),u_{m}(x)$ and $u_{{\it\tau}}(x)$ . In § 4 the flow field will be described in detail for the case that these four functions are given.

4. Flow field

4.1. Asymptotic solution far downstream

As already mentioned in the introduction, the turbulent wall-jet flow will be considered as a perturbation of the turbulent half-free jet flow. Turbulent free shear flows are slender, i.e. the mean local lateral velocity component is small compared to the dominant mean flow velocity. The flow spreads gradually, thus axial gradients are small in comparison to lateral gradients. These features allow the application of boundary-layer equations in place of the full Navier–Stokes equations (see Pope Reference Pope2000, p. 111). Since the turbulent half-free jet flow is surrounded by quiescent fluid, the continuity equation and the momentum equation for the $x$ -direction are

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle u\;\frac{\partial u}{\partial x}+v\;\frac{\partial u}{\partial y}=\frac{\partial }{\partial y}\left(\frac{{\it\tau}_{t}}{{\it\rho}}\right), & \displaystyle\end{eqnarray}$$

where ${\it\tau}_{t}$ is the turbulent shear stress. It is assumed that the velocity distribution of the turbulent half-free jet is self-similar:

(4.3) $$\begin{eqnarray}u=U_{N}(x)\;{\dot{F}}(\tilde{{\it\eta}}).\end{eqnarray}$$

In (4.3) the similarity variable is

(4.4) $$\begin{eqnarray}\tilde{{\it\eta}}=k\;\frac{y}{y_{0.5}(x)}.\end{eqnarray}$$

The dot in (4.3) refers to differentiation with respect to $\tilde{{\it\eta}}$ . The constant $k$ is defined by

(4.5) $$\begin{eqnarray}{\dot{F}}(k)={\textstyle \frac{1}{2}}.\end{eqnarray}$$

From the continuity equation (4.1) it follows that

(4.6) $$\begin{eqnarray}v=-\frac{1}{k}\;\frac{\text{d}(U_{N}y_{0.5})}{\text{d}x}\;F+\frac{U_{N}}{k}\;\frac{\text{d}y_{0.5}}{\text{d}x}\;\tilde{{\it\eta}}\;{\dot{F}}.\end{eqnarray}$$

The momentum equation (4.2) leads to the shear stress:

(4.7) $$\begin{eqnarray}\frac{{\it\tau}_{t}}{{\it\rho}}=\frac{y_{0.5}}{k}\;U_{N}\;\frac{\text{d}U_{N}}{\text{d}x}\;F\;{\dot{F}}+\frac{2}{3}\;\frac{U_{N}}{k}\;\left(2y_{0.5}\;\frac{\text{d}U_{N}}{\text{d}x}+U_{N}\;\frac{\text{d}y_{0.5}}{\text{d}x}\right)\;F^{3}.\end{eqnarray}$$

Dimensional analysis requires that $y_{0.5}$ is proportional to $x-x_{0}$ , where $x_{0}$ is the coordinate of the virtual origin of the half-free jet:

(4.8) $$\begin{eqnarray}y_{0.5}=4{\it\alpha}k(x-x_{0}).\end{eqnarray}$$

The rate of spreading of the half-free jet is $4{\it\alpha}k$ , and ${\it\alpha}$ is a slenderness parameter.

As will be mentioned later, several turbulence models applied to turbulent free jets can be found in the literature. In the turbulence model used here it is assumed that the eddy viscosity ${\it\nu}_{t}$ is independent of $\tilde{{\it\eta}}$ (see Pope Reference Pope2000, p. 137).

Dimensional analysis then yields the ansatz

(4.9) $$\begin{eqnarray}{\it\nu}_{t}(x)=\frac{{\it\alpha}}{k}\;U_{N}(x)\;y_{0.5}(x).\end{eqnarray}$$

Therefore it follows for the shear stress that

(4.10) $$\begin{eqnarray}\frac{{\it\tau}_{t}}{{\it\rho}}=k\;\frac{{\it\nu}_{t}}{y_{0.5}}\;\frac{\partial u}{\partial \tilde{{\it\eta}}}={\it\alpha}\;U_{N}^{2}(x)\;\ddot{F}(\tilde{{\it\eta}}).\end{eqnarray}$$

Combination of (4.7) and (4.10) leads to a differential equation for the function $F(\tilde{{\it\eta}})$ . With $U_{N}(x)\sim (x-x_{0})^{-1/2}$ the following ordinary differential equation results:

(4.11) $$\begin{eqnarray}\ddot{F}+2\;F\;{\dot{F}}=0,\end{eqnarray}$$

because the factor in front of $F^{3}$ in (4.7) vanishes.

The boundary conditions for $F(\tilde{{\it\eta}})$ are

(4.12) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\tilde{{\it\eta}}=0:~F=0,\quad {\dot{F}}=1,\quad \ddot{F}=0\\ \tilde{{\it\eta}}\rightarrow \infty :~F=1,\quad {\dot{F}}=0,\quad \ddot{F}=0.\end{array}\right\}\end{eqnarray}$$

The solution of (4.11) is

(4.13a,b ) $$\begin{eqnarray}F=\tanh \tilde{{\it\eta}},\quad {\dot{F}}=1-\left(\tanh \tilde{{\it\eta}}\right)^{2}.\end{eqnarray}$$

A comparison of this solution with experimental results will be discussed later in § 4.5.

The momentum flux

(4.14) $$\begin{eqnarray}K(x)=\int _{0}^{\infty }u^{2}(x,y)\,\text{d}y\end{eqnarray}$$

can now be determined for the turbulent half-free jet:

(4.15) $$\begin{eqnarray}K_{\infty }={\textstyle \frac{8}{3}}{\it\alpha}\;(x-x_{0})\;U_{N}^{2}(x),\end{eqnarray}$$

which is independent of $x$ . This leads to the formula

(4.16) $$\begin{eqnarray}U_{N}(x)=\sqrt{\frac{3\;K_{\infty }}{8{\it\alpha}\;(x-x_{0})}}.\end{eqnarray}$$

The turbulence model expressed by (4.9) has been chosen here because of the simple solution for the velocity distribution ${\dot{F}}(\tilde{{\it\eta}})$ . Solutions for turbulent free jet flows with alternative turbulence modelling can also be found in the literature, for example Tollmien (Reference Tollmien1926), Saffman (Reference Saffman1970), Schneider & Mörwald (Reference Schneider and Mörwald1987) and Schneider (Reference Schneider1991). These solutions lead to different differential equations for $F(\tilde{{\it\eta}})$ and, hence, to different constants in (4.16). However, the hypothesis $K_{\infty }\neq 0$ proposed here would not be affected by the choice of the turbulence model for the half-free jet flow.

4.2. Viscous wall layer

In this case results for the viscous wall layer of turbulent boundary layers can be adopted, see Schlichting & Gersten (Reference Schlichting and Gersten2003, pp. 572, 523). The $x$ -component of the velocity is given by the universal distribution

(4.17) $$\begin{eqnarray}u^{+}(y^{+})=\frac{u(x,y)}{u_{{\it\tau}}(x)}=F(y^{+})\end{eqnarray}$$

where

(4.18) $$\begin{eqnarray}y^{+}=\frac{y\,u_{{\it\tau}}}{{\it\nu}}\end{eqnarray}$$

and

(4.19) $$\begin{eqnarray}u_{{\it\tau}}(x)=\sqrt{\frac{{\it\tau}_{w}}{{\it\rho}}}.\end{eqnarray}$$

The limiting form of the function $F(y^{+})$ is

(4.20) $$\begin{eqnarray}\lim _{y^{+}\rightarrow \;\infty }\;u^{+}(y^{+})=\frac{1}{{\it\kappa}}\ln y^{+}+C^{+}\end{eqnarray}$$

where ${\it\kappa}$ is called the Kármán constant and $C^{+}$ is a further universal constant, determined from numerous experiments to be $C^{+}=5.0$ for smooth walls. In the literature, (4.20) is denoted as the ‘logarithmic law of the wall’. For high Reynolds numbers the $y$ -component of the velocity vanishes in the viscous wall layer, and the shear stress ${\it\tau}(x)$ is independent of $y^{+}$ and, hence, equal to the wall shear stress ${\it\tau}_{w}(x)$ , see Schlichting & Gersten (Reference Schlichting and Gersten2003, p. 573). An analytic description of the universal function $F(y^{+})$ is given in Schlichting & Gersten (Reference Schlichting and Gersten2003, p. 524).

4.3. Overlap layer

The wall layer $(y\leqslant y_{m})$ of turbulent wall jets has the same three-layer structure as turbulent boundary layers. These layers are the viscous wall layer, the so-called defect layer $(0\leqslant y\leqslant y_{m})$ and the overlap layer in between. The latter is part of the viscous wall layer and hence independent of $y_{m}$ , as well as part of the defect layer and hence independent of the kinematic viscosity ${\it\nu}$ . Therefore the gradient of the velocity $u^{+}$ in the overlap layer is independent of ${\it\nu}$ and $y_{m}$ .

Hence, the matching condition for the velocity gradient reads as follows:

(4.21) $$\begin{eqnarray}\lim _{{\it\eta}\rightarrow 0}{\it\eta}\frac{\partial u^{+}}{\partial {\it\eta}}=\lim _{y^{+}\rightarrow \infty }y^{+}\frac{\partial u^{+}}{\partial y^{+}}=\frac{1}{{\it\kappa}},\end{eqnarray}$$

where

(4.22) $$\begin{eqnarray}{\it\eta}=\frac{y}{y_{m}}\end{eqnarray}$$

is the dimensionless distance from the wall. Integrating the wall-layer part in (4.21) leads to (4.20). Integrating the defect-layer part in (4.21) leads to

(4.23) $$\begin{eqnarray}\lim _{{\it\eta}\rightarrow 0}\;u^{+}({\it\eta})=\frac{u_{m}}{u_{{\it\tau}}}+\frac{1}{{\it\kappa}}\ln {\it\eta}-\bar{C},\end{eqnarray}$$

where the integration constant is defined as

(4.24) $$\begin{eqnarray}\bar{C}=\lim _{{\it\eta}\rightarrow 0}\;\int _{{\it\eta}}^{1}\left(\frac{\text{d}u^{+}}{\text{d}{\it\eta}}-\frac{1}{{\it\kappa}{\it\eta}}\right)\,\text{d}{\it\eta}.\end{eqnarray}$$

The matching condition for the velocity $u^{+}$ in the overlap layer reads as follows:

(4.25) $$\begin{eqnarray}\frac{u_{m}}{u_{{\it\tau}}}-\bar{C}+\frac{1}{{\it\kappa}}\ln {\it\eta}=\frac{1}{{\it\kappa}}\ln y^{+}+C^{+}\end{eqnarray}$$

or

(4.26) $$\begin{eqnarray}\frac{1}{{\it\gamma}}=\frac{1}{{\it\kappa}}\ln \left({\it\gamma}\;\mathit{Re}_{m}\right)+C^{+}+\bar{C}\end{eqnarray}$$

with

(4.27a,b ) $$\begin{eqnarray}{\it\gamma}=\frac{u_{{\it\tau}}}{u_{m}}=\sqrt{\frac{c_{f}}{2}},\quad \mathit{Re}_{m}=\frac{u_{m}\;y_{m}}{{\it\nu}}.\end{eqnarray}$$

This implicit friction law $c_{f}=f(\mathit{Re}_{m})$ can be written in explicit form as

(4.28) $$\begin{eqnarray}\sqrt{\frac{c_{f}}{2}}={\it\gamma}=\frac{{\it\kappa}}{\ln \mathit{Re}}_{m}\;G({\it\Lambda}_{m};\;D_{m},\;E_{m})\end{eqnarray}$$

where

(4.29) $$\begin{eqnarray}\displaystyle & {\it\Lambda}_{m}=2\ln \mathit{Re}_{m} & \displaystyle\end{eqnarray}$$

(4.30) $$\begin{eqnarray}\displaystyle & D_{m}=2\;[\ln (2{\it\kappa})+{\it\kappa}\;(C^{+}+\bar{C})]=2.036 & \displaystyle\end{eqnarray}$$

(4.31) $$\begin{eqnarray}\displaystyle & E_{m}=0. & \displaystyle\end{eqnarray}$$

The function $G({\it\Lambda};D,E)$ has been defined and tabulated by Gersten & Herwig (Reference Gersten and Herwig1992). More details are given in appendix A. The friction law according to (4.28) is shown in figure 2. The agreement with experimental data by Launder & Rodi (Reference Launder and Rodi1981) and Hammond (Reference Hammond1982) is very good. As will become apparent later, the limits $\mathit{Re}_{m}\rightarrow \infty$ and ${\it\gamma}\rightarrow 0$ are equivalent to the limits $x\rightarrow \infty$ and $\mathit{Re}_{x}\rightarrow \infty$ , respectively.

The dotted line in figure 2 will be discussed in § 9.

4.4. Defect layer $(0<y\leqslant y_{m})$

4.4.1. Flow field

In the limiting case of infinite Reynolds number (which is equivalent to $y_{m}/y_{0.5}\rightarrow 0$ ), the velocity $u(x,y)$ in the defect layer attains the value $u_{m}(x)$ . Therefore it seems natural to write the velocity in the form of a defect law

(4.32) $$\begin{eqnarray}u(x,{\it\eta})=u_{m}(x)-u_{{\it\tau}}(x)\;f^{\prime }({\it\eta})\end{eqnarray}$$

with

(4.33) $$\begin{eqnarray}{\it\eta}=\frac{y}{y_{m}}.\end{eqnarray}$$

The continuity equation (4.1) leads to

(4.34) $$\begin{eqnarray}v(x,{\it\eta})=\frac{\text{d}y_{m}}{\text{d}x}\;u_{{\it\tau}}\left(f-{\it\eta}\;f^{\prime }\right)+y_{m}\;\frac{\text{d}u_{{\it\tau}}}{\text{d}x}\;f-y_{m}\;\frac{\text{d}u_{m}}{\text{d}x}\;{\it\eta}.\end{eqnarray}$$

When the velocity components are known, the shear stress can be determined by using the momentum equation (4.2). Transforming (4.2) into the $x,{\it\eta}$ coordinate system and using (4.32) leads to

(4.35) $$\begin{eqnarray}y_{m}\;u_{m}\;\frac{\text{d}u_{m}}{\text{d}x}+u_{{\it\tau}}^{2}\,\left(A{\it\eta}f^{\prime \prime }+B\;f^{\prime }+C\;f^{\prime 2}+D\;f\;f^{\prime \prime }\right)=\frac{\partial }{\partial {\it\eta}}\left(\frac{{\it\tau}_{t}}{{\it\rho}}\right)\end{eqnarray}$$

where

(4.36a−d ) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle A(x)=\frac{1}{u_{{\it\tau}}}\;\frac{\text{d}\left(y_{m}\;u_{m}\right)}{\text{d}x},\quad B(x)=-\frac{y_{m}}{u_{{\it\tau}}^{2}}\;\frac{\text{d}(u_{{\it\tau}}\;u_{m})\;}{\text{d}x},\\ \displaystyle C(x)=\frac{y_{m}}{u_{{\it\tau}}}\;\frac{\text{d}u_{{\it\tau}}}{\text{d}x},\quad D(x)=-\frac{1}{u_{{\it\tau}}}\;\frac{\text{d}(u_{{\it\tau}}\;y_{m})}{\text{d}x},\end{array}\right\},\end{eqnarray}$$

see Schlichting & Gersten (Reference Schlichting and Gersten2003, p. 574).

Splitting the shear stress into two parts

(4.37) $$\begin{eqnarray}\frac{{\it\tau}_{t}(x,{\it\eta})}{{\it\rho}}=y_{m}\;u_{m}\;\frac{\text{d}u_{m}}{\text{d}x}{\it\eta}+u_{{\it\tau}}^{2}\;s(x,{\it\eta})\end{eqnarray}$$

reduces the momentum equation (4.35) to

(4.38) $$\begin{eqnarray}A{\it\eta}f^{\prime \prime }+B\;f^{\prime }+C\;f^{\prime 2}+D\;f\;f^{\prime \prime }=s^{\prime }.\end{eqnarray}$$

The term $s^{\prime }$ refers to differentiating $s(x,{\it\eta})$ with respect to ${\it\eta}$ .

Using the condition $s(x,0)=1$ , integration of (4.38) leads to

(4.39) $$\begin{eqnarray}s(x,{\it\eta})=1+A\left({\it\eta}\;f^{\prime }-f\right)+B\;f+C\int _{0}^{{\it\eta}}f^{\prime 2}\,\text{d}{\it\eta}+D\int _{0}^{{\it\eta}}f\,f^{\prime \prime }\,\text{d}{\it\eta}.\end{eqnarray}$$

4.4.2. Determination of the function $f({\it\eta})$

The function $f({\it\eta})$ represents the self-similarity of the velocity distribution in the defect layer according to (4.32). This function must satisfy the differential equation (4.38), which in this case must be independent of $x$ . This means that $A$ and $B$ in (4.38) must be constants, whereas $C$ and $D$ will become negligibly small and $s({\it\eta})$ depends only on ${\it\eta}$ .

Now one should keep in mind that the defect layer of turbulent wall jets behaves like the defect layer of turbulent equilibrium boundary layers. The latter are defined by the self-similarity of the velocity distribution, see Schlichting & Gersten (Reference Schlichting and Gersten2003, p. 580). For turbulent equilibrium boundary layers the functions $A(x)$ and $B(x)$ are constant for high Reynolds numbers. It is therefore appropriate to assume the analogous behaviour for the wall jet in the limit $\mathit{Re}_{x}\rightarrow \infty$ :

(4.40a,b ) $$\begin{eqnarray}A(\mathit{Re}_{x}\rightarrow \infty )=A_{\infty }\quad B(\mathit{Re}_{x}\rightarrow \infty )=B_{\infty },\end{eqnarray}$$

which will be verified in (4.47) after the following lines of formulae.

Introducing the characteristic length

(4.41) $$\begin{eqnarray}\tilde{{\rm\Delta}}(x)=\frac{u_{m}\;y_{m}}{u_{{\it\tau}}}\end{eqnarray}$$

leads to

(4.42a−d ) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle A(x)=\frac{\text{d}\tilde{{\rm\Delta}}}{\text{d}x}+\frac{\tilde{{\rm\Delta}}}{u_{{\it\tau}}}\;\frac{\text{d}u_{{\it\tau}}}{\text{d}x},\quad B(x)=-\left(\frac{\tilde{{\rm\Delta}}}{u_{m}}\;\frac{\text{d}u_{m}}{\text{d}x}+\frac{\tilde{{\rm\Delta}}}{u_{{\it\tau}}}\;\frac{\text{d}u_{{\it\tau}}}{\text{d}x}\right)\\ \displaystyle C(x)={\it\gamma}\;\frac{\tilde{{\rm\Delta}}}{u_{{\it\tau}}}\;\frac{\text{d}u_{{\it\tau}}}{\text{d}x},\quad D(x)=-{\it\gamma}\;\left(A(x)+\frac{\tilde{{\rm\Delta}}}{{\it\gamma}}\;\frac{\text{d}{\it\gamma}}{\text{d}x}\right).\end{array}\right\}\end{eqnarray}$$

From the definition of ${\it\gamma}$ in (4.27a ) it follows that

(4.43) $$\begin{eqnarray}\frac{\tilde{{\rm\Delta}}}{u_{{\it\tau}}}\;\frac{\text{d}u_{{\it\tau}}}{\text{d}x}=\frac{\tilde{{\rm\Delta}}}{u_{m}}\;\frac{\text{d}u_{m}}{\text{d}x}+\frac{\tilde{{\rm\Delta}}}{{\it\gamma}}\;\frac{\text{d}{\it\gamma}}{\text{d}x}.\end{eqnarray}$$

Differentiation of (4.26) with respect to $x$ leads to

(4.44) $$\begin{eqnarray}\frac{\tilde{{\rm\Delta}}}{{\it\gamma}}\;\frac{\text{d}{\it\gamma}}{\text{d}x}=-\frac{A(x)}{{\it\kappa}}\;{\it\gamma}+O({\it\gamma}^{2}).\end{eqnarray}$$

For the limit ${\it\gamma}\rightarrow 0$ the terms proportional to $\text{d}{\it\gamma}/\text{d}x$ in (4.42a−d ) and (4.43) are small compared to the other terms and can be neglected. In the flow far downstream $u_{m}(x)$ changes into $U_{N}(x)$ according to (4.16). From (4.42b ) and (4.43) it follows that

(4.45) $$\begin{eqnarray}\frac{\tilde{{\rm\Delta}}}{u_{m}}\;\frac{\text{d}u_{m}}{\text{d}x}=\frac{\tilde{{\rm\Delta}}}{U_{N}}\;\frac{\text{d}U_{N}}{\text{d}x}=-\frac{\tilde{{\rm\Delta}}(x)}{2\;\left(x-x_{0}\right)}=-\frac{B_{\infty }}{2}.\end{eqnarray}$$

Therefore $\tilde{{\rm\Delta}}(x)$ is proportional to $x-x_{0}$ or to $y_{0.5}(x)$ according to (4.8):

(4.46) $$\begin{eqnarray}\tilde{{\rm\Delta}}(x)=A_{1}\;y_{0.5}(x)=4{\it\alpha}kA_{1}\;(x-x_{0}).\end{eqnarray}$$

The limiting constant $A_{\infty }=A(x\rightarrow \infty )$ reads as follows:

(4.47) $$\begin{eqnarray}A_{\infty }=\frac{B_{\infty }}{2}=2{\it\alpha}kA_{1}\end{eqnarray}$$

and, hence, equation (4.44) reduces to

(4.48) $$\begin{eqnarray}\frac{\text{d}{\it\gamma}}{\text{d}x}=-\frac{{\it\gamma}^{2}}{2{\it\kappa}\;(x-x_{0})}+O({\it\gamma}^{3}),\end{eqnarray}$$

which after integration leads to

(4.49) $$\begin{eqnarray}{\it\gamma}=\frac{{\it\kappa}}{\ln \mathit{Re}_{x}}\;[1+o(1)],\end{eqnarray}$$

where $\mathit{Re}_{x}$ is defined in (2.3). This formula shows clearly that the limit $\mathit{Re}_{x}\rightarrow \infty$ is equivalent to the limit ${\it\gamma}\rightarrow 0$ . Combining (4.41) and (4.46) leads to

(4.50) $$\begin{eqnarray}\frac{y_{m}}{y_{0.5}}=A_{1}\;\frac{u_{{\it\tau}}}{u_{m}}=A_{1}\;{\it\gamma}.\end{eqnarray}$$

The ratio $y_{m}/y_{0.5}$ tends to zero far downstream, as has been anticipated in the present hypothesis for $K_{\infty }\neq 0$ .

In § 6 equation (4.50) will be extended to an asymptotic series expansion (6.1). By using this extension it can be shown that the functions $A(x)$ and $B(x)$ have for ${\it\gamma}\rightarrow 0$ the following asymptotic form:

(4.51a,b ) $$\begin{eqnarray}A(x)=A_{\infty }+O({\it\gamma}),\quad B(x)=B_{\infty }+O({\it\gamma}).\end{eqnarray}$$

The function $f({\it\eta})$ has to satisfy the following ordinary differential equation, which results from (4.39) for the limit $x\rightarrow \infty$ :

(4.52) $$\begin{eqnarray}s({\it\eta})=1+A_{\infty }\;({\it\eta}\;f^{\prime }-f)+B_{\infty }\;f.\end{eqnarray}$$

In order to enable the solution of the ordinary differential equation (4.52), a turbulence model is needed that provides a relation between $f({\it\eta})$ and $s({\it\eta})$ . Instead of that, an indirect turbulence model will be applied here. A plausible function $f({\it\eta})$ will be chosen and the corresponding function $s({\it\eta})$ can then be determined from (4.52).

The function $f({\it\eta})$ must satisfy the following boundary conditions:

(4.53) $$\begin{eqnarray}\displaystyle & {\it\eta}=1:~f=f_{m},\quad f^{\prime }=0,\quad f^{\prime \prime }=0,\quad f^{\prime \prime \prime }=0 & \displaystyle\end{eqnarray}$$

(4.54) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\eta}\rightarrow 0:~f=0,\quad f^{\prime }=\bar{C}-\frac{1}{{\it\kappa}}\ln {\it\eta},\quad f^{\prime \prime }=-\frac{1}{{\it\kappa}{\it\eta}}. & \displaystyle\end{eqnarray}$$

For ${\it\eta}\rightarrow 0$ the condition $f=0$ follows from (4.34) for $v(x,{\it\eta}\rightarrow 0)=0$ ; the conditions for $f^{\prime }$ and $f^{\prime \prime }$ have been derived in (4.23).

In particular the physical significance of the integration constant $\bar{C}$ is apparent in (4.24). The condition $f^{\prime \prime \prime }({\it\eta}=1)=0$ results from ‘patching’ the term $\partial ^{2}u/\partial y^{2}$ of the defect layer and the outer layer at $y=y_{m}$ (see (4.63)).

For ${\it\eta}=1$ the value $f_{m}$ is a measure of $v(x,{\it\eta}=1)$ , and the conditions $f^{\prime }=0$ and $f^{\prime \prime }=0$ result from $u=u_{m}$ .

The following function consisting of a logarithmic term and a power series satisfies all boundary conditions appearing in (4.53) and (4.54):

(4.55) $$\begin{eqnarray}f^{\prime }({\it\eta})=\frac{1}{{\it\kappa}}\left(-\text{ln}\,{\it\eta}-\frac{5}{6}+\frac{3}{2}{\it\eta}^{2}-\frac{2}{3}{\it\eta}^{3}\right)\end{eqnarray}$$

with

(4.56a−c ) $$\begin{eqnarray}{\it\kappa}=0.41,\quad \bar{C}=-\frac{5}{6{\it\kappa}}=-2.033,\quad f_{m}=\frac{1}{2{\it\kappa}}=1.220.\end{eqnarray}$$

In figure 3 the defect velocity

(4.57) $$\begin{eqnarray}\frac{u_{m}-u}{u_{{\it\tau}}}=f^{\prime }({\it\eta})\end{eqnarray}$$

is compared with experimental data of Wygnanski et al. (Reference Wygnanski, Katz and Horev1992). The difference between theory and experiment may be attributed to the fact that the experiments refer to fairly low Reynolds numbers. The maximum of the absolute difference for $f^{\prime }({\it\eta})=5$ is about ${\rm\Delta}y=0.2~\text{mm}$ , which is surprisingly small. The theoretical curve $f^{\prime }({\it\eta})$ does not depend on any empirical constant. It is proportional to $1/{\it\kappa}$ according to (4.55), where ${\it\kappa}$ is the well-defined ((4.21), $\mathit{Re}\rightarrow \infty$ , zero pressure gradient) universal Kármán constant ${\it\kappa}=0.41$ .

Figure 3.

Self-similar distribution of the defect velocity in the defect layer.

4.5. Outer layer $(y\geqslant y_{m})$

It is assumed that the distribution of the velocity is again self-similar:

(4.58) $$\begin{eqnarray}u=u_{m}(x)\;{\dot{F}}(\bar{{\it\eta}})\end{eqnarray}$$

where the similarity variable is

(4.59) $$\begin{eqnarray}\bar{{\it\eta}}=\frac{y-y_{m}(x)}{{\rm\Delta}(x)}\end{eqnarray}$$

with

(4.60) $$\begin{eqnarray}{\rm\Delta}(x)=\frac{y_{0.5}-y_{m}}{k}.\end{eqnarray}$$

The dot designates differentiation with respect to $\bar{{\it\eta}}$ . The variable $\bar{{\it\eta}}$ is a generalization of the variable $\tilde{{\it\eta}}$ in (4.4) for $y_{m}\neq 0$ .

From the continuity equation it follows that

(4.61) $$\begin{eqnarray}v=-\frac{\text{d}(u_{m}{\rm\Delta})}{\text{d}x}\;F+u_{m}\;\frac{\text{d}y_{m}}{\text{d}x}\;{\dot{F}}+u_{m}\;\frac{\text{d}{\rm\Delta}}{\text{d}x}\;\bar{{\it\eta}}\;{\dot{F}}+\frac{\text{d}(y_{m}\;u_{{\it\tau}})}{\text{d}x}\;f_{m}-\frac{\text{d}(y_{m}\;u_{m})}{\text{d}x}.\end{eqnarray}$$

The momentum equation leads to the shear stress:

(4.62) $$\begin{eqnarray}\displaystyle \frac{{\it\tau}_{t}}{{\it\rho}} & = & \displaystyle u_{m}{\rm\Delta}\;\frac{\text{d}u_{m}}{\text{d}x}\;F\;{\dot{F}}+\frac{2}{3}\;u_{m}\;\left(2{\rm\Delta}\;\frac{\text{d}u_{m}}{\text{d}x}+u_{m}\;\frac{\text{d}{\rm\Delta}}{\text{d}x}\right)\;F^{3}\nonumber\\ \displaystyle & & \displaystyle +\,\left[\frac{\text{d}(y_{m}\;u_{{\it\tau}})}{\text{d}x}\;f_{m}-\frac{\text{d}(y_{m}\;u_{m})}{\text{d}x}\right]\;u_{m}\;{\dot{F}}+u_{{\it\tau}}^{2}+\frac{\text{d}(y_{m}\;u_{m}^{2}\;)}{\text{d}x}\nonumber\\ \displaystyle & & \displaystyle -\,\left[\;u_{{\it\tau}}\;\frac{\text{d}(y_{m}\;u_{m})}{\text{d}x}+y_{m}\;\frac{\text{d}(u_{{\it\tau}}\;u_{m})}{\text{d}x}+u_{m}\;\frac{\text{d}(y_{m}\;u_{{\it\tau}})}{\text{d}x}\right]\;f_{m}.\end{eqnarray}$$

Equations (4.61) and (4.62) reduce to (4.6) and (4.7) for the limit $x\rightarrow \infty$ , when $y_{m}\rightarrow 0$ , $u_{m}\rightarrow U_{N}(x)$ and ${\rm\Delta}\rightarrow y_{0.5}/k$ are taken into account.

At $y=y_{m}$ the values for $u$ , $\partial u/\partial y$ , $v$ and ${\it\tau}_{t}/{\it\rho}$ for the defect layer $({\it\eta}=1)$ and the outer layer $(\bar{{\it\eta}}=0)$ are equal (‘patching’). For $\partial ^{2}u/\partial y^{2}$ the patching condition is as follows:

(4.63) $$\begin{eqnarray}f^{\prime \prime \prime }({\it\eta}=1)=2\left[\frac{k\;A_{1}}{1-A_{1}\;{\it\gamma}}\right]^{2}{\it\gamma}.\end{eqnarray}$$

In the limit ${\it\gamma}\rightarrow 0$ it follows that $f^{\prime \prime \prime }(1)=0$ in (4.53).

The function $F(\bar{{\it\eta}})$ is identical with $F(\tilde{{\it\eta}})$ . All results for $F(\tilde{{\it\eta}})$ in § 4.1 can be used in this generalization. Figure 4 shows the comparison of the solution for the outer layer with experimental results of Tailland & Mathieu (Reference Tailland and Mathieu1967).

Figure 4.

Self-similar velocity distribution in the outer layer.

4.6. Determination of the momentum flux $K(x)$

The momentum flux

(4.64) $$\begin{eqnarray}K(x)=\int _{0}^{\infty }u^{2}(x,y)\,\text{d}y\end{eqnarray}$$

can now be determined if (4.32) and (4.58) are used for the integration. The result is

(4.65) $$\begin{eqnarray}K(x)=0.756\;u_{m}^{2}\;y_{0.5}\left(1+0.322\;\frac{y_{m}}{y_{0.5}}-2.644\;f_{m}\;\sqrt{\frac{c_{f}}{2}}\;\frac{y_{m}}{y_{0.5}}\right).\end{eqnarray}$$

The constants are

(4.66a−d ) $$\begin{eqnarray}0.756=\frac{2}{3k},\quad 0.322=\frac{3k}{2}-1,\quad 2.644=3k,\quad k=0.8814.\end{eqnarray}$$

The formula (4.65) can be used to determine the momentum flux $K$ when the global values in the formula are known.

5. Momentum-integral equation

The momentum flux $K(x)$ must satisfy the momentum-integral equation:

(5.1) $$\begin{eqnarray}K(x)=K_{\infty }+\int _{x}^{\infty }\frac{{\it\tau}_{w}}{{\it\rho}}\,\text{d}x.\end{eqnarray}$$

A dimensionless friction velocity

(5.2) $$\begin{eqnarray}{\it\gamma}_{G}(x)=\frac{u_{{\it\tau}}(x)}{U_{N}(x)}\end{eqnarray}$$

is introduced, where $U_{N}(x)$ is given in (4.16). Then (5.1) can be written as

(5.3) $$\begin{eqnarray}\frac{K(x)}{K_{\infty }}=1+\frac{3}{8{\it\alpha}}\;\int _{x}^{\infty }\frac{{\it\gamma}_{G}^{2}(x)\,\text{d}x}{x-x_{o}}=1+\frac{3}{4{\it\alpha}}\;\int _{{\it\Lambda}}^{\infty }{\it\gamma}_{G}^{2}({\it\Lambda})\,\text{d}{\it\Lambda},\end{eqnarray}$$

where

(5.4) $$\begin{eqnarray}{\it\Lambda}=\ln \mathit{Re}_{x}=\ln \left(\sqrt{K_{\infty }(x-x_{o})}/{\it\nu}\right).\end{eqnarray}$$

The integration leads to the final formula (see appendix A):

(5.5) $$\begin{eqnarray}\frac{K({\it\Lambda})}{K_{\infty }}=1+\frac{3{\it\kappa}^{2}}{4{\it\alpha}}\left\{\frac{{\it\gamma}_{G}({\it\Lambda})}{{\it\kappa}}+\left[\frac{{\it\gamma}_{G}({\it\Lambda})}{{\it\kappa}}\right]^{2}+O\left(\frac{1}{{\it\Lambda}^{3}}\right)\right\}.\end{eqnarray}$$

(This equation without the term $O({\it\gamma}_{G}^{2})$ appears also in Gersten & Herwig (Reference Gersten and Herwig1992) and in Schlichting & Gersten (Reference Schlichting and Gersten2003). Unfortunately, there the Greek letters ${\it\kappa}$ and ${\it\alpha}$ got lost in the printing process.)

6. Wall-jet flow described by perturbation method

So far the flow field of the wall jet has been described for given functions $y_{m}(x),y_{0.5}(x),u_{m}(x)$ and $u_{{\it\tau}}(x)$ . These functions are known in the limiting case $x\rightarrow \infty$ , see § 4.1. Now, for the general case, these functions will be described by a perturbation method. From (5.5) it is obvious to use ${\it\gamma}_{G}$ as the perturbation parameter. The following power series expansions will be applied (with respect to $A_{1}$ see (4.50)):

(6.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{y_{m}}{y_{0.5}}=A_{1}\;{\it\gamma}_{G}\left(1+A_{2}\;{\it\gamma}_{G}+\cdots \right) & \displaystyle\end{eqnarray}$$

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{m}}{U_{N}}=1+B_{1}\;{\it\gamma}_{G}+B_{2}\;{\it\gamma}_{G}^{2}+\cdots \,. & \displaystyle\end{eqnarray}$$

It is assumed that the asymptotic formula (see (4.8))

(6.3) $$\begin{eqnarray}y_{0.5}=4{\it\alpha}k(x-x_{0})\end{eqnarray}$$

is generally valid, which is in accordance with experiments. Introducing (6.1) and (6.2) into (4.27) yields

(6.4a,b ) $$\begin{eqnarray}{\it\gamma}={\it\gamma}_{G}\;\frac{U_{N}}{u_{m}},\quad \mathit{Re}_{m}=\frac{u_{m}}{U_{N}}\;\frac{y_{m}}{y_{0.5}}\;P\;\mathit{Re}_{x},\end{eqnarray}$$

where

(6.5) $$\begin{eqnarray}P=k\;\sqrt{6{\it\alpha}}.\end{eqnarray}$$

The skin friction law (4.26) is now transformed into

(6.6) $$\begin{eqnarray}\frac{1}{{\it\gamma}_{G}}=\frac{1}{{\it\kappa}}\ln \left({\it\gamma}_{G}^{2}\;\mathit{Re}_{x}\right)+\hat{D}+\hat{E}\;{\it\gamma}_{G}\end{eqnarray}$$

with

(6.7a,b ) $$\begin{eqnarray}\hat{D}=\frac{1}{{\it\kappa}}\ln (A_{1}\;P)-B_{1}+C^{+}+\bar{C},\quad \hat{E}=\frac{1}{{\it\kappa}}\;A_{2}-B_{2}.\end{eqnarray}$$

In appendix A it is shown that the explicit solution of (6.6) is

(6.8) $$\begin{eqnarray}{\it\gamma}_{G}=\frac{{\it\kappa}}{\ln \mathit{Re}_{x}}\;G({\it\Lambda};\;D,\;E)\end{eqnarray}$$

where

(6.9a−c ) $$\begin{eqnarray}{\it\Lambda}=\ln \mathit{Re}_{x},\quad D=2\ln {\it\kappa}+{\it\kappa}\;\hat{D},\quad E={\it\kappa}^{2}\;\hat{E}.\end{eqnarray}$$

7. Determination of the universal constants

It is desirable to determine the universal constants ${\it\alpha}$ , $A_{1}$ , $A_{2}$ , $B_{1}$ and $B_{2}$ defined in (4.8), (6.1) and (6.2) in such a way that the analysis is optimally concordant with existing experiments. Unfortunately, all existing experimental data on plane turbulent wall jets belong to rather low Reynolds numbers and, hence, are far away from the asymptotic state. Despite this deficiency, matching of the analysis with experimental data will be attempted.

Experimental results on six available turbulent wall-jet flows have been analysed. The main global values are listed in table 1 below. The constant ${\it\alpha}~(=\,\text{d}y_{0.5}/\text{d}x/(4k))$ introduced in (4.8) can be taken from Launder & Rodi (Reference Launder and Rodi1981), who analysed about 20 experimental investigations. They found ${\it\alpha}=0.021$ . This is in accordance with the mean value resulting from the experiments in the table ( $k$ from (4.66d )):

(7.1a,b ) $$\begin{eqnarray}{\it\alpha}=0.021,\quad P=k\;\sqrt{6{\it\alpha}}=0.3129.\end{eqnarray}$$

Taking into account (4.16), (6.1) and (6.2) and comparing (5.5) and (4.65) provide the following two conditions that the constants $A_{1}$ , $A_{2}$ , $B_{1}$ and $B_{2}$ must satisfy:

(7.2) $$\begin{eqnarray}\displaystyle & \displaystyle 0.322\;A_{1}+2\;B_{1}=\frac{3{\it\kappa}}{4{\it\alpha}} & \displaystyle\end{eqnarray}$$

(7.3) $$\begin{eqnarray}\displaystyle & \displaystyle A_{1}\;(0.644\;B_{1}-2.644\;f_{m})+B_{1}^{2}+2\;B_{2}+0.322\;A_{1}\;A_{2}=\frac{3}{4{\it\alpha}}. & \displaystyle\end{eqnarray}$$

Table 1.

Experimental data.

Two still missing relations between the constants follow from the condition that the experimental values of two particular wall-jet velocity profiles should coincide with the corresponding theoretical values. The following two points have been selected for the agreement with the theory:

Point I (from Tailland & Mathieu 3, $x-x_{0}=1.06~\text{m}$ )

(7.4a,b ) $$\begin{eqnarray}\frac{y_{m}}{y_{0.5}}=0.155,\quad \frac{u_{m}\;y_{0.5}}{{\it\nu}}=0.447\times 10^{5}.\end{eqnarray}$$

Point II (from Tailland & Mathieu 1, $x-x_{0}=1.05~\text{m}$ )

(7.5a,b ) $$\begin{eqnarray}\frac{y_{m}}{y_{0.5}}=0.131,\quad \frac{u_{m}\;y_{0.5}}{{\it\nu}}=1.003\times 10^{5}.\end{eqnarray}$$

In addition to (7.2) and (7.3), the (6.1) and (6.2) specified for each of these points lead finally to six nonlinear algebraic equations for $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , ${\it\gamma}_{GI}$ and ${\it\gamma}_{GII}$ . The results are the universal constants:

(7.6a−d ) $$\begin{eqnarray}A_{1}=1.4798,\quad A_{2}=1.7441,\quad B_{1}=7.1002,\quad B_{2}=-8.7227.\end{eqnarray}$$

The positions of the points are given by:

(7.7) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\it\gamma}_{GI}=0.08923~(\mathit{Re}_{x}=0.9091\times 10^{5},~\mathit{Re}_{m}=6789)\\ {\it\gamma}_{GII}=0.07944~(\mathit{Re}_{x}=2.1287\times 10^{5},~\mathit{Re}_{m}=13\,451).\end{array}\right\}\end{eqnarray}$$

The two selected points I and II are marked in figures 2 and 5 by triangles.

Figure 5.

Universal functions. $K/K_{\infty }$ according to (5.5) and (6.6),  $u_{m}\;y_{0.5}/{\it\nu}=\mathit{Re}_{m}\;y_{0.5}/y_{m}$ , $\mathit{Re}_{m}$ and ${\it\gamma}$ according to (6.4), ${\it\gamma}_{G}$ according to (6.6). Experiments: see table 1 for symbols. ${\rm\Delta}$ denotes selected points for agreement with theory.

The numerical values of the universal constants should be considered preliminary until experimental data for higher Reynolds numbers become available.

8. Determination of the asymptotic momentum flux $K_{\infty }$

When the values $u_{m}$ , $y_{0.5}$ and ${\it\nu}$ at the position $x-x_{0}$ are known, the asymptotic momentum flux $K_{\infty }$ can be determined by using the universal function shown in figure 5. The combination $u_{m}\;y_{0.5}/{\it\nu}$ leads to the corresponding value $\mathit{Re}_{x}$ and, via (2.3), to $K_{\infty }$ . All positions $x$ in a particular wall jet should lead to the same value $K_{\infty }$ . This would be true to a greater extent the further downstream the positions are. The mean value $\bar{K}_{\infty }$ of all analysed profiles of a particular wall jet is considered as the asymptotic momentum flux of this wall jet.

When the values $K$ and ${\it\nu}$ at the position $x-x_{0}$ are known $K_{\infty }$ can be determined by using the universal function for $K(\mathit{Re}_{x})/K_{\infty }$ , also shown in figure 5. Here an iteration process would lead to the solution $K_{\infty }$ .

9. Comparisons with experiments

Six turbulent wall jets documented in the literature have been analysed. The $\bar{K}_{\infty }$ -values and other characteristic parameters are listed in table 1. The values $u_{m}\;y_{0.5}/{\it\nu}$ of the analysed wall jets are in good agreement with the theoretical universal curve in figure 5. The experimental results by Bradshaw & Gee (Reference Bradshaw and Gee1960) are not shown in figure 5 because they are practically identical with the results by Tailland & Mathieu 3 (Reference Tailland and Mathieu1967). The universal relation between $K/\bar{K}_{\infty }$ and $\mathit{Re}_{x}$ is also shown in figure 5.

The experimental data for the thickness ratio $y_{m}/y_{0.5}$ are given in table 1 for four wall jets. They are compared with the present theory in figure 2. The short bars refer to the values of $y_{m}/y_{0.5}$ according to the table averaged over the covered range of Reynolds numbers $\mathit{Re}_{m}$ . The authors obviously assumed that the thickness ratio is a constant for the entire wall jet. The analysis of this paper, however, shows that $y_{m}/y_{0.5}$ decreases monotonically with increasing $\mathit{Re}_{m}$ . The dotted line in figure 2 corresponds to (6.1). It is a necessary condition for the hypothesis $K_{\infty }\neq 0$ that $y_{m}/y_{0.5}$ tends to zero for $\mathit{Re}_{m}\rightarrow \infty$ (see $u/u_{m}$ in figure 6).

Figure 6.

Distributions of velocity and shear stress at point II. ——  ${\it\gamma}_{G}=0.0794$ , $\mathit{Re}_{x}=2.1\times 10^{5}$ , after (9.1)–(9.4), ○ experiment, after Tailland & Mathieu (Reference Tailland and Mathieu1967).

The ratio of $K_{\infty }/K_{j}$ is not a universal constant, but it varies in the table between 0.19 and 0.45. It depends mainly on the velocity distribution at the inflow. Narasimha et al. (Reference Narasimha, Narayan and Parthasarathy1973) have non-dimensionalized $x-x_{0}$ by $K_{j}$ rather than by $K_{\infty }$ as has been done in the present investigation. Asymptotic formulae should not depend on $K_{j}$ .

In figure 6 the theoretical distributions of the axial velocity and the shear stress are compared with experimental data of Tailland & Mathieu (Reference Tailland and Mathieu1967). The corresponding formulae (valid for all turbulent wall jets) are

(9.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{u}{u_{m}}=1-({\it\gamma}_{G}-7.10\;{\it\gamma}_{G}^{2})\;f^{\prime }({\it\eta})\quad {\it\eta}=\frac{y}{y_{m}}\leqslant 1 & \displaystyle\end{eqnarray}$$

(9.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{u}{u_{m}}={\dot{F}}(\bar{{\it\eta}})\quad \bar{{\it\eta}}=0.8814\;\frac{y-y_{m}}{y_{0.5}-y_{m}}\geqslant 0 & \displaystyle\end{eqnarray}$$

(9.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\tau}_{t}}{{\it\rho}\;U_{N}^{2}}=-0.0548{\it\eta}{\it\gamma}_{G}+[1+0.0548\;({\it\eta}\;f^{\prime }+f)-0.8734{\it\eta}]{\it\gamma}_{G}^{2}\quad 0<{\it\eta}\leqslant 1 & \displaystyle\end{eqnarray}$$

(9.4) $$\begin{eqnarray}\displaystyle \frac{{\it\tau}_{t}}{{\it\rho}\;U_{N}^{2}} & = & \displaystyle 0.021\;\ddot{F}+[0.534\;\ddot{F}-0.0548\;{\dot{F}}]\;{\it\gamma}_{G}\nonumber\\ \displaystyle & & \displaystyle +\,[0.757\;\ddot{F}-0.866\;(F^{3}-1)-0.673\;{\dot{F}}]\;{\it\gamma}_{G}^{2}\quad \bar{{\it\eta}}\geqslant 0.\end{eqnarray}$$

It should be emphasized that the limiting solution does not satisfy the no-slip condition at the wall and has zero wall shear stress. The solution including the first-order disturbances satisfies the no-slip condition, but still has zero wall shear stress. Only if the second-order disturbances are taken into account does the shear stress have a finite value at the wall.

In figure 6 the special case ${\it\gamma}_{G}=0.0794$ is plotted and compared with the limiting curve ${\it\gamma}_{G}=0$ . Obviously, the comparisons between the analytical results and the experimental data in figures 2–6 make the hypothesis $K_{\infty }\neq 0$ plausible. A definitely convincing verification, however, will be possible in the future, preferably by direct numerical solutions of turbulent wall jets.