3.3.1 Mean velocity

Figure 11(a) shows mean streamwise velocity profiles at $x/h=25$ , 30 and 35. The profiles are scaled with the outer parameters $y_{1/2}$ and $U_{max}$ . For the given range in the streamwise direction, the profiles show self-similar behaviour. Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) showed that the mean streamwise velocity profiles show self-similar behaviour in outer scales beyond $x/h=20$ . The mean flow predicted by previous LES (Banyassady & Piomelli Reference Banyassady and Piomelli2014) is in agreement with the DNS. The current results also compare well with the measurements of Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) and Eriksson et al. (Reference Eriksson, Karlsson and Persson1998). This is with the exception of close to the edge of outer layer beyond $y/y_{1/2}>1.8$ . This is due to the difference in outer flow conditions. The experiments have reverse flow for entrainment.

In the presence of a weak co-flow with the velocity $U_{\infty }$ , the outer scaling for the mean streamwise velocity is defined as $(\langle u\rangle -U_{\infty })/(U_{max}-U_{\infty })$ (Irwin Reference Irwin1973; Zhou et al. Reference Zhou, Heine and Wygnanski1996) and $y_{1/2}$ is located where $\langle u\rangle =(1/2)(U_{max}-U_{\infty })$ . Figure 11(b) shows that the mean streamwise velocity profiles at $x/h=25$ , $30$ and $35$ collapse with this scaling in the outer layer region. The velocity profiles are compared with the experimental data of Irwin (Reference Irwin1973) at $Re_{j}=28\,000$ with $U_{\infty }/U_{max}=0.38$ .

Figure 12 shows the outer-scaled mean wall-normal velocity $\langle v\rangle$ . The velocity profiles do not collapse beyond $y/y_{1/2}=1.2$ . The calculated wall-normal velocity is in good agreement with the measurements of Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) up to $y/y_{1/2}=1.6$ . The difference in measurements and computed values in the outer region is again due to different entrainment conditions.

Figure 12. Mean wall-normal velocity profiles scaled with outer parameters. Experimental data: Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) (▵), $x=70h$ .

Figure 13. Inner-scaled mean streamwise velocity profiles $\langle u\rangle ^{+}$ . Log-law $\langle u\rangle ^{+}=2.44\ln (y^{+})+5.0$ (— ⋅ ⋅ —). Linear profile $\langle u\rangle ^{+}=y^{+}$ ( $\cdots \cdots$ ). LES of Banyassady & Piomelli (Reference Banyassady and Piomelli2014) (●), $x=30h$ . Experimental data: Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) (○), $x=30h$ ; Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) (▵), $x=40h$ .

Figure 13 shows the inner-scaled mean streamwise velocity $\langle u\rangle ^{+}$ profiles in a semi-logarithmic form. The computed profiles at three streamwise locations $x/h=25$ , 30 and 45 collapse up to $y^{+}=300$ . The current DNS is in agreement with the experimental data (Eriksson et al. Reference Eriksson, Karlsson and Persson1998; Rostamy et al. Reference Rostamy, Bergstrom, Sumner and Bugg2011a ). The LES of Banyassady & Piomelli (Reference Banyassady and Piomelli2014) gives slightly lower values for $\langle u\rangle ^{+}$ in the range of $10\leqslant y^{+}\leqslant 370$ , than the current DNS. There is agreement with the linear profile $\langle u\rangle ^{+}=y^{+}$ below $y^{+}=4$ , which is similar to the flat plate turbulent boundary layer (Wu & Moin Reference Wu and Moin2009). The near wall velocity profiles for wall bounded flows are also expressed as a log law of the form $\langle u\rangle ^{+}=A\ln (y^{+})+B$ . The current DNS is compared with a log law having $A=2.44$ and $B=5.0$ . These constants are the generally accepted values for flat plate turbulent boundary layers (Spalart Reference Spalart1988). Several previous experimental measurements of wall jets (Eriksson et al. Reference Eriksson, Karlsson and Persson1998; Tachie et al. Reference Tachie, Balachandar and Bergstrom2004; Rostamy et al. Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) have shown agreement with these log-law parameters. Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) showed that their measurements are in agreement with the log law in the range of $30\leqslant y^{+}\leqslant 80$ , whereas the current DNS range is $20\leqslant y^{+}\leqslant 90$ . The log-law parameters for the LES (Banyassady & Piomelli Reference Banyassady and Piomelli2014) are $A=2.22$ and $B=5.0$ .

Figure 14. Mean streamwise velocity profiles scaled with incomplete similarity parameters of Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005): (a) outer scaled profiles and (b) inner-scaled profiles.

Figure 14 represents mean streamwise velocity profiles with incomplete similarity parameters described by Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005). The outer scaling parameters for incomplete similarity are traditionally defined except for the factor $A_{o}$ used for scaling the $y$ -axis (figure 14 a). The parameter $A_{o}$ is given in (3.4), which describes the dependence of the outer layer length scale $y_{1/2}$ on the streamwise distance $x/h$ . The semi-logarithmic plot clearly shows that outer scale mean streamwise velocity profiles above $y_{max}$ show perfect collapse, whereas profiles below $y_{max}$ diverge. Figure 14(b) shows the inner-scaled profiles of mean streamwise velocity. The $y$ -axis is scaled with $A_{i}$ and $(y_{1/2})_{in}$ given in (3.5). The inner scaling is able to collapse the velocity profile in the inner layer region below $y_{max}$ . Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005) gave a relationship for mean streamwise velocity based on incomplete similarity as

(3.8) $$\begin{eqnarray}\displaystyle \langle u\rangle =\left\{\begin{array}{@{}ll@{}}\displaystyle \left(\frac{M}{\unicode[STIX]{x1D70C}h}\right)^{1/2}\left(\frac{x}{h}\right)^{\unicode[STIX]{x1D6FE}_{m}}\unicode[STIX]{x1D713}_{u}\left(\frac{y}{h^{1-\unicode[STIX]{x1D6FE}_{o}}x^{\unicode[STIX]{x1D6FE}_{o}}},Re\right),\quad & \text{if }y>y_{max};\\ \displaystyle \left(\frac{M}{\unicode[STIX]{x1D70C}h}\right)^{1/2}\left(\frac{x}{h}\right)^{\unicode[STIX]{x1D6FE}_{m}}\unicode[STIX]{x1D713}_{u}\left(\frac{y}{h^{1-\unicode[STIX]{x1D6FE}_{i}}x^{\unicode[STIX]{x1D6FE}_{i}}},Re\right),\quad & \text{if }y<y_{max}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D713}_{u}$ is a function of length scales and Reynolds number. Earlier it was shown that the current DNS and several recent measurements give $\unicode[STIX]{x1D6FE}_{m}$ close to $-0.5$ , which results in a weak dependence on inlet slot height and hence at high Reynolds number complete similarity is possible. It is important to point out that even if the hypothesis of incomplete similarity is not applicable for the wall jet, the inner layer parameters of (3.5) suggested by Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005) show the same quality of scaling as the inner scaling suggested by George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000), based on the asymptotic invariance principle (AIP). It can be shown that the parameters given in (3.5) are consistent with the similarity theory of George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000). The detailed derivation is given in George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000), here the essential relationships are referenced to check the scaling parameters. It has been shown (George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000) that at infinite Reynolds number the momentum equation in the inner layer region is given as

(3.9) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\left[\langle -u^{\prime }v^{\prime }\rangle +\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}y}\right]=0, & & \displaystyle\end{eqnarray}$$

where $\langle u\rangle =0$ at $y=0$ . Equation (3.9) can be integrated to obtain

(3.10) $$\begin{eqnarray}\displaystyle \langle -u^{\prime }v^{\prime }\rangle +\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}y}=\frac{\unicode[STIX]{x1D70F}_{w}}{\unicode[STIX]{x1D70C}}=u_{\unicode[STIX]{x1D70F}}. & & \displaystyle\end{eqnarray}$$

The form of similarity solution is given as

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \langle u\rangle =U_{si}(x)f_{i\infty }(y_{si}), & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \langle -u^{\prime }v^{\prime }\rangle =R_{si}(x)r_{i\infty }(y_{si}), & \displaystyle\end{eqnarray}$$

where $U_{si}$ , $f_{i\infty }$ , $R_{si}$ and $r_{i\infty }$ are spatial functions. $y_{si}=y/l_{si}$ and length scale $l_{si}=l_{si}(x)$ is required for proper scaling. Substituting these solutions in (3.10) gives

(3.13) $$\begin{eqnarray}\displaystyle \left[\frac{u_{\unicode[STIX]{x1D70F}}^{2}}{U_{si}^{2}}\right]=\left[\frac{R_{si}}{U_{si}^{2}}\right]r_{i\infty }+\left[\frac{\unicode[STIX]{x1D708}}{l_{si}U_{si}}\right]f_{i\infty }^{\prime }. & & \displaystyle\end{eqnarray}$$

For the similarity solution all the bracketed terms should have same $x$ -dependence i.e.

(3.14) $$\begin{eqnarray}\displaystyle \left[\frac{u_{\unicode[STIX]{x1D70F}}^{2}}{U_{si}^{2}}\right]\sim \left[\frac{R_{si}}{U_{si}^{2}}\right]\sim \left[\frac{\unicode[STIX]{x1D708}}{l_{si}U_{si}}\right]. & & \displaystyle\end{eqnarray}$$

It has been shown by George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000) that if the length scale is defined as $l_{si}=\unicode[STIX]{x1D708}/U_{si}$ , $u_{\unicode[STIX]{x1D70F}}$ must be the inner velocity scale. Now if we use inner length scale defined in (3.5) i.e. $l_{si}=(y_{1/2})_{in}$ , then any appropriate velocity can be used for the scaling. In this case $U_{max}$ is the obvious choice and figure 14(b) clearly shows that these parameters are appropriate scales in the inner layer region. The only other requirement from the George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000) similarity theory is that the Reynolds shear stress should also scale with $U_{max}$ and $(y_{1/2})_{in}$ in the inner layer region, which is shown in the next section.

3.3.2 Reynolds stresses

Figure 15 shows outer-scaled Reynolds normal and shear stress profiles at streamwise locations $x/h=25$ , 30 and 35. The normal and shear stresses are normalized by $(U_{max}-U_{\infty })^{2}$ . The streamwise $\langle u^{\prime }u^{\prime }\rangle$ , wall-normal $\langle v^{\prime }v^{\prime }\rangle$ and shear stresses $\langle u^{\prime }v^{\prime }\rangle$ from the current DNS are compared with the measurements of Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) at $Re_{j}=9600$ , Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) at $Re_{j}=7500$ and the LES of Banyassady & Piomelli (Reference Banyassady and Piomelli2014) also at $Re_{j}=7500$ . The DNS results are close to the reported LES and slightly higher than the measurements of Eriksson et al. (Reference Eriksson, Karlsson and Persson1998). These measurements are at a higher Reynolds number than the current DNS. Note, Wygnanski et al. (Reference Wygnanski, Katz and Horev1992) showed that with increasing Reynolds number, outer-scaled values of $\langle u^{\prime }u^{\prime }\rangle$ decrease slightly. The experimental data of Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) give higher values for all the stresses at the same Reynolds number as the current DNS. It is important to note here that the current DNS relies solely on transition and resolution of the production mechanism for turbulence generation. On the other hand, the LES of Banyassady & Piomelli (Reference Banyassady and Piomelli2014) used forcing at streamwise locations $x/h=2$ , 4, 6 and 8 in the wall-normal momentum equation. This gave Reynolds shear stress profiles equal to the measurements of Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) at these locations. Beyond the forcing planes, the LES allowed the flow to evolve naturally, however, even with this forcing, predicted stresses are still lower than the measurements. A possible reason for the higher values of Reynolds normal and shear stresses of Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) might be the uncertainties in the measurement of their scaling parameters. Figure 15(c) shows the outer-scaled Reynolds stress profiles in the spanwise direction $\langle w^{\prime }w^{\prime }\rangle$ . Few experimental studies have measured $\langle w^{\prime }w^{\prime }\rangle$ , however, the current DNS shows agreement with the measurements of Irwin (Reference Irwin1973) at $Re_{j}=28\,000$ .

Figure 15. Outer-scaled Reynolds normal and shear stress profiles: (a) streamwise $\langle u^{\prime }u^{\prime }\rangle$ ; (b) wall-normal $\langle v^{\prime }v^{\prime }\rangle$ ; (c) spanwise $\langle w^{\prime }w^{\prime }\rangle$ and (d) shear stress $\langle u^{\prime }v^{\prime }\rangle$ . LES of Banyassady & Piomelli (Reference Banyassady and Piomelli2014) (●), $x=30h$ . Experimental data: Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) (○), $x=30h$ ; Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) (▵), $x=40h$ ; Irwin (Reference Irwin1973) (▿), $x=82.2h$ .

It has been shown by several experimental studies (Irwin Reference Irwin1973; Abrahamsson, Johansson & Löfdahl Reference Abrahamsson, Johansson and Löfdahl1994; Zhou et al. Reference Zhou, Heine and Wygnanski1996; Eriksson et al. Reference Eriksson, Karlsson and Persson1998; Rostamy et al. Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) that Reynolds normal and shear stress profiles exhibit self-similar behaviour with outer scaling in the developed region of the wall jet. The Reynolds stresses have shown self-similarity as early as $x/h=30$ (Rostamy et al. Reference Rostamy, Bergstrom, Sumner and Bugg2011a ). The outer-scaled Reynolds normal and shear stress profiles from the current DNS in figure 15 do not show the same level of collapse as the mean streamwise velocity profiles in figures 11 or 14. However, the maximum differences in the peak values for these DNS profiles at $x/h=30$ and 35 are less than the experimental uncertainty given by Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ).

Figure 16. Inner-scaled Reynolds normal and shear stress profiles (a) streamwise $\langle u^{\prime }u^{\prime }\rangle ^{+}$ , (b) wall-normal $\langle v^{\prime }v^{\prime }\rangle ^{+}$ , (c) spanwise $\langle w^{\prime }w^{\prime }\rangle ^{+}$ and (d) shear stress $\langle u^{\prime }v^{\prime }\rangle ^{+}$ and velocity gradient $y^{+}(\text{d}\langle u\rangle ^{+}/\text{d}y^{+})$ at $x=30h$ . LES of Banyassady & Piomelli (Reference Banyassady and Piomelli2014) (●), $x=30h$ . Experimental data: Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) (○), $x=30h$ ; Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) (▵), $x=40h$ .

Figure 16 shows the inner-scaled Reynolds normal and shear stress profiles. The friction velocity $u_{\unicode[STIX]{x1D70F}}$ and inner length scale $\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ are the velocity and length scales. Again, the streamwise $\langle u^{\prime }u^{\prime }\rangle ^{+}$ , wall-normal $\langle v^{\prime }v^{\prime }\rangle ^{+}$ and shear stress $\langle u^{\prime }v^{\prime }\rangle ^{+}$ profiles from the current DNS are compared with the experimental data of Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) and Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) and the LES of Banyassady & Piomelli (Reference Banyassady and Piomelli2014). The measurements of Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) and LES are close to the current DNS for $\langle v^{\prime }v^{\prime }\rangle ^{+}$ and $\langle u^{\prime }v^{\prime }\rangle ^{+}$ and lower for $\langle u^{\prime }u^{\prime }\rangle ^{+}$ in the inner layer region. The inner layer region extends up to $y/y_{1/2}=0.2$ , or $y^{+}=160$ . The measurements of Rostamy et al. (Reference Rostamy, Bergstrom, Sumner and Bugg2011a ) are significantly higher than the current DNS for $\langle u^{\prime }u^{\prime }\rangle ^{+}$ and $\langle v^{\prime }v^{\prime }\rangle ^{+}$ , whereas $\langle u^{\prime }v^{\prime }\rangle ^{+}$ is in agreement. The $\langle v^{\prime }v^{\prime }\rangle ^{+}$ and $\langle u^{\prime }v^{\prime }\rangle ^{+}$ profiles at $x/h=25$ , 30 and 35 collapse in the inner layer region. Whereas $\langle u^{\prime }u^{\prime }\rangle ^{+}$ and $\langle w^{\prime }w^{\prime }\rangle ^{+}$ have a small variation, this is less than the uncertainty levels in the measurements (Rostamy et al. Reference Rostamy, Bergstrom, Sumner and Bugg2011a ). Figure 16(d) also compares the velocity gradient profile $y^{+}(\text{d}\langle u\rangle ^{+}/\text{d}y^{+})$ at $x=30h$ , with the Reynolds shear stress. The velocity gradient becomes zero at $y=y_{max}$ or $y^{+}=161$ , where the Reynolds shear stress has a finite positive value. Moreover, for a narrow region below $y=y_{max}$ both the velocity gradient and Reynolds shear stress are positive. This invalidates the Boussinesq hypothesis $\langle u^{\prime }v^{\prime }\rangle ^{=}-\unicode[STIX]{x1D708}_{T}(\unicode[STIX]{x2202}\langle u\rangle /\unicode[STIX]{x2202}y)$ for the wall jet, where the positive scalar coefficient $\unicode[STIX]{x1D708}_{T}$ is the turbulent viscosity. It also shows that the positive shear stress from the outer layer is transported against the velocity gradient below $y=y_{max}$ due to the turbulence transport.

Figure 17. (a) Correct scaling of shear stress profile according to George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000). Experimental data: Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) (▵), $x=40h$ and (b) inner-scaled shear stress profiles with respect to incomplete similarity parameters (Barenblatt et al. Reference Barenblatt, Chorin and Prostokishin2005).

George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000) showed, using the asymptotic invariance principle (AIP), that for correct outer scaling, the shear stress $\langle u^{\prime }v^{\prime }\rangle$ should be normalized with the shear velocity $u_{\unicode[STIX]{x1D70F}}^{2}$ . Figure 17(a) shows the shear stress profiles with this scaling. The profiles at $x/h=30$ and 35 show an improvement in their collapse with respect to this new scaling (George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000) relative to the scaling based on a single velocity and length scale (Irwin Reference Irwin1973). This can be seen in figure 15(d). The current-scaled profiles are in agreement with the experimental data of Eriksson et al. (Reference Eriksson, Karlsson and Persson1998) up to $y/y_{1/2}=0.8$ . Beyond this, experimental values for $\langle u^{\prime }v^{\prime }\rangle ^{+}$ are higher. This might be due to the higher Reynolds number for the experiment and difference in wall friction velocity.

Figure 17(b) shows the shear stress profiles scaled with incomplete similarity parameters (Barenblatt et al. Reference Barenblatt, Chorin and Prostokishin2005). It has been discussed earlier that the incomplete similarity parameters for the inner layer region would be consistent with the asymptotic invariance principle (George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Lofdahal and Wosnik2000), if they scale the Reynolds shear stress. The figure shows a good collapse of shear stress profiles with this scaling in the inner layer region.

Figure 18. Frequency spectra of velocity and pressure fluctuations at $x/h=30$ (a) streamwise $E_{u^{\prime }}$ at: $y^{+}=5$ (– – –); $y^{+}=17$ (— ⋅ —); $y/y_{1/2}=0.2$ (——); $y/y_{1/2}=0.8$ (— ⋅ ⋅ —), (b) wall-normal $E_{v^{\prime }}$ at: $y^{+}=5$ (– – –); $y^{+}=17$ (— ⋅ —); $y/y_{1/2}=0.2$ (——); $y_{1/2}=0.8$ (— ⋅ ⋅ —), (c) spanwise $E_{w^{\prime }}$ at: $y^{+}=5$ (– – –); $y^{+}=40$ (— ⋅ —); $y/y_{1/2}=0.2$ (——); $y/y_{1/2}=0.8$ (— ⋅ ⋅ —) and (d) pressure $E_{p^{\prime }}$ at: $y^{+}=5$ (– – –); $y^{+}=17$ (— ⋅ —); $y/y_{1/2}=0.2$ (——); $y/y_{1/2}=0.8$ (— ⋅ ⋅ —).

The instantaneous values of velocity and pressure are saved at selected wall-normal locations at $x/h=30$ , for each time step from $t^{\ast }=1200$ to 2500, giving 860 000 samples. Note, $St=fh/U_{j}$ is the Strouhal number or non-dimensional frequency and $f$ is the frequency. As shown in figure 18, the streamwise $E_{u^{\prime }}$ and wall-normal $E_{v^{\prime }}$ spectra are given at $y^{+}=5$   $(y/y_{1/2}=0.006)$ , $y^{+}=17$ $(y/y_{1/2}=0.02)$ , $y/y_{1/2}=0.2$ and $y/y_{1/2}=0.8$ . These locations represent the end of the linear region in the viscous sub layer, the first peak in $\langle u^{\prime }u^{\prime }\rangle$ , $U_{max}$ and the outer layer peaks in $\langle u^{\prime }u^{\prime }\rangle$ and $\langle v^{\prime }v^{\prime }\rangle$ , respectively. The Reynolds stress $\langle u^{\prime }u^{\prime }\rangle$ has a lower value at $y^{+}=5$ , and contains less energy at the smaller scales relative to the other locations (figure 18 a). At the other three locations $\langle u^{\prime }u^{\prime }\rangle$ values are close to each other and so too are the spectra. The wall-normal Reynolds stress $\langle v^{\prime }v^{\prime }\rangle$ increases continuously from the wall to a peak value around $y/y_{1/2}=0.8$ . Correspondingly spectra at increasing wall distance indicate a higher energy level. The spanwise velocity fluctuation spectra $E_{w^{\prime }}$ (figure 18 c) are given at the same locations as $E_{u^{\prime }}$ and $E_{v^{\prime }}$ , except for $y^{+}=40$ $(y/y_{1/2}=0.05)$ , which is the first near wall peak in $\langle w^{\prime }w^{\prime }\rangle$ . The streamwise velocity fluctuation spectra show a $-5/3$ slope in the range of $0.06<St<2.0$ in the outer layer region. The wall-normal and spanwise velocity fluctuations spectra have a $-5/3$ slope in a smaller range of frequencies $0.4<St<2.0$ in the outer layer region. The higher frequency region is the viscous sub-range, where dissipation occurs and spectra can be compared to a line with a slope of $-7$ . The spectra in the inner layer region are closer to such a line as compared to the outer layer region.

In the low frequency region below $St<0.06$ spectra, particularly for $E_{u^{\prime }}$ and $E_{v^{\prime }}$ , peaks indicating large scale fluctuations in the flow can be observed. Figure 18(d) shows the pressure spectra $E_{p^{\prime }}$ at identical locations to the spectra for $E_{u^{\prime }}$ and $E_{v^{\prime }}$ . The main features in the pressure spectra are multiple peaks in the low frequency range for $St<0.06$ , as observed for $E_{u^{\prime }}$ and $E_{v^{\prime }}$ . The peaks appear to be the signature of large scale structures passing in the outer layer region identified in figure 5.

3.4.1 Inner-scaled budgets

Figure 19 shows the budgets for $tke$ , Reynolds normal and shear stresses at $x/h=30$ in the inner layer region. The profiles are scaled with the inner variables, whereas the budget terms are normalized with $u_{\unicode[STIX]{x1D70F}}^{4}/\unicode[STIX]{x1D708}$ . The balance or the sum of all the budget terms for each Reynolds stress is $O(10^{-2})$ of the maximum value. As mentioned earlier, the current budgets are compared with the wall jet LES of Dejoan & Leschziner (Reference Dejoan and Leschziner2005). This LES might be the only published budget for comparison for wall jets in the inner layer region. The LES based budget is given at $x/h=20$ and only the dominant terms from that budget are included here. The dominant terms of the turbulence kinetic energy and Reynolds stress budgets from the flat plate turbulent boundary layer DNS of Spalart (Reference Spalart1988) are also included in figure 19. This helps to understand how closely the inner layer of the plane wall jet follows a turbulent boundary layer flow.

It is clearly shown that (figure 19 a,b,d,e) the dominant terms of production, dissipation and velocity–pressure gradient correlation for turbulent kinetic energy, Reynolds streamwise, spanwise and shear stress budgets for the wall jet and boundary layer are in agreement. The wall-normal Reynolds stress budget shows (figure 19 c) that the dominant terms of dissipation and velocity–pressure gradient correlation have different peak values from the boundary layer, however the trend is the same for both flows. The turbulent transport term for the wall-normal and shear stress budgets indicate a major deviation of the wall jet from the turbulent boundary layer. This has a significant effect on the Reynolds wall-normal and shear stress distribution of the wall jet.

In figure 19(a) a comparison of the $tke$ budget with previously reported LES (Dejoan & Leschziner Reference Dejoan and Leschziner2005) shows that the level of production is lower than the current DNS and the trend for the dissipation does not match below $y^{+}=20$ . The Reynolds wall-normal and shear stress budgets (figure 19 c,e) from LES show even more drastic deviation from the current DNS. In the case of Reynolds shear stress the LES gives a significantly lower level of production and velocity–pressure gradient correlation. The wall-normal stress from the LES does not follow standard wall behaviour, rather than velocity–pressure gradient correlation term it gives turbulent transport as the dominant term, which is balanced by the dissipation. Moreover, the trend for turbulent transport and level of dissipation for wall-normal stress from LES do not match with the current DNS. A possible explanation is that the LES predicted these budgets at $x/h=20$ , where the wall jet boundary layer may not be fully developed and the outer shear layer is interacting with the wall. However, the current DNS shows that the velocity–pressure gradient term is dominant for the $\langle v^{\prime }v^{\prime }\rangle$ budget in the inner layer region as far upstream as $x/h=15$ . Also, the dissipation term in the LES is not calculated explicitly, but evaluated as a balance from the rest of the terms. This might be responsible for the near wall difference below $y^{+}=20$ in $\langle u^{\prime }u^{\prime }\rangle$ , where sub-grid modelling may have some deficiencies. The overall difference between the current DNS and LES is due to the lower grid resolution of the latter, as mentioned earlier.

For the DNS, as can be seen from figure 19(a) in the inner layer region, the turbulent kinetic energy budget shows that the dissipation is balanced by viscous diffusion in the viscous sub-layer for $y^{+}<5$ . The production term has high values outside the viscous sub-layer in the range of $5\leqslant y^{+}\leqslant 50$ , with a peak around $y^{+}=12$ . This high value of production is balanced mainly by dissipation and up to a certain extent, by turbulence diffusion. Eriksson (Reference Eriksson2003) estimated the near wall dissipation value as 0.27, which is 12 % lower than the current value of 0.31.

Figure 19(b) shows the streamwise Reynolds stress $\langle u^{\prime }u^{\prime }\rangle$ budget. This is similar to the turbulence kinetic energy budget, except for the velocity–pressure gradient correlation term. This changes the sign and balances production along with dissipation and turbulent diffusion. The velocity–pressure gradient correlation transfers streamwise energy to other directions. The high level of production in the region of $5\leqslant y^{+}\leqslant 50$ is responsible for the inner layer peak of the streamwise Reynolds stress (figure 16 a). For the wall-normal stress $\langle v^{\prime }v^{\prime }\rangle$ budget in the inner layer region (figure 19 c), the dissipation is mainly balanced with the velocity–pressure gradient correlation and turbulent diffusion terms. The production is small and wall-normal turbulence in the inner layer region is maintained by turbulent diffusion and velocity–pressure gradient correlation. These transfer turbulence energy from the streamwise to the wall-normal direction. Figure 19(d) shows the budget for the spanwise Reynolds stress $\langle w^{\prime }w^{\prime }\rangle$ . The dissipation is balanced by viscous diffusion in the viscous sub-layer region $y^{+}<5$ . Outside the viscous sub-layer, it is balanced with the velocity–pressure gradient term. The Reynolds shear stress $\langle u^{\prime }v^{\prime }\rangle$ budget (figure 19 e) has high negative production, which is balanced by the velocity–pressure gradient term and turbulent diffusion.

Figure 20. Turbulence kinetic energy ( $tke$ ), Reynolds normal and shear stress budgets in the outer layer region. The terms are normalized with $(U_{max}-U_{\infty })^{3}/y_{1/2}$ . Legends for DNS are same as in figure 19. (a) $tke=(\langle u^{\prime }u^{\prime }\rangle +\langle v^{\prime }v^{\prime }\rangle +\langle w^{\prime }w^{\prime }\rangle )/2$ , (b) $\langle u^{\prime }u^{\prime }\rangle$ , (c) $\langle v^{\prime }v^{\prime }\rangle$ , (d) $\langle w^{\prime }w^{\prime }\rangle$ and (e) $\langle u^{\prime }v^{\prime }\rangle$ . LES of wall jet by Dejoan & Leschziner (Reference Dejoan and Leschziner2005): production (▴); dissipation (▪); velocity–pressure gradient correlation (●). Experimental data, Irwin (Reference Irwin1973): production (▵); dissipation (▫); turbulent diffusion ( $+$ ) and Zhou et al. (Reference Zhou, Heine and Wygnanski1996): Production (▿); dissipation (♢); turbulent diffusion ( $\ast$ ).

3.4.2 Outer-scaled budgets

Figure 20 shows the turbulence kinetic energy, Reynolds normal and shear stress budgets at $x/h=30$ in the outer layer region. The budget terms are normalized with $(U_{max}-U_{\infty })^{3}$ and wall-normal distance with $y_{1/2}$ . The balance for the outer-scaled budgets is less than 4 % of the peak values of production and dissipation. The outer scale LES budgets of Dejoan & Leschziner (Reference Dejoan and Leschziner2005) are compared with the current DNS. The predicted turbulence kinetic energy budget is also compared with the measurements of Irwin (Reference Irwin1973) and Zhou et al. (Reference Zhou, Heine and Wygnanski1996).

Figure 20(a) shows the outer-scaled turbulent kinetic energy budget, where all the terms have been evaluated explicitly for the current DNS. The viscous diffusion and velocity–pressure gradient terms are negligible in the outer layer region. The production and convection terms are mainly balanced by turbulent diffusion and dissipation. The production term has a minimum around $y_{max}$ , however it always remains positive. The production, dissipation and turbulent transport terms are compared with the measurements. In the experiments only turbulent transport can be measured directly and the DNS values lie between the two sets of measurements (Irwin Reference Irwin1973; Zhou et al. Reference Zhou, Heine and Wygnanski1996) and is closer to Irwin’s data. In experiments the production is estimated from a mean curve drawn through the measured mean velocity values (Irwin Reference Irwin1973). Both experiments cited here give identical values of production and are close to the current DNS. The dissipation is estimated either from local spectra using the $-5/3$ law (Irwin Reference Irwin1973) or using the assumption of local isotropy along with Taylor’s hypothesis (Zhou et al. Reference Zhou, Heine and Wygnanski1996). The dissipation estimates from Irwin (Reference Irwin1973) are close to the current DNS, whereas Zhou et al. (Reference Zhou, Heine and Wygnanski1996) have estimated a higher level of dissipation. The current DNS shows that the assumption of isotropy in dissipation is not valid in the inner layer region below $y/y_{1/2}=0.2=y_{max}$ . The DNS shows that in the outer layer region for $0.2\leqslant y/y_{1/2}\leqslant 1$ the wall-normal dissipation $\langle (\unicode[STIX]{x2202}v^{\prime }/\unicode[STIX]{x2202}x_{k})(\unicode[STIX]{x2202}v^{\prime }/\unicode[STIX]{x2202}x_{k})\rangle$ is 15–20 % smaller and the spanwise dissipation $\langle (\unicode[STIX]{x2202}w^{\prime }/\unicode[STIX]{x2202}x_{k})(\unicode[STIX]{x2202}w^{\prime }/\unicode[STIX]{x2202}x_{k})\rangle$ is 10–15 % smaller than the streamwise dissipation $\langle (\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}x_{k})(\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}x_{k})\rangle$ , respectively. The dominant terms of production and velocity–pressure gradient correlation from LES (Dejoan & Leschziner Reference Dejoan and Leschziner2005) are slightly lower than the current DNS, whereas the dissipation is in good agreement for various budgets.

Figure 20(b) shows the outer-scaled streamwise Reynolds stress $\langle u^{\prime }u^{\prime }\rangle$ budget. The production has high positive values in the range of $0.2<y/y_{1/2}<1.5$ , which is responsible for high values of Reynolds stress $\langle u^{\prime }u^{\prime }\rangle$ in the outer layer region. A portion of this energy is dissipated and the remainder transfers to turbulent and velocity–pressure gradient diffusion. The turbulent production has a minimum value at $y_{max}$ , where maximum mean velocity occurs and $\unicode[STIX]{x2202}\langle u\rangle /\unicode[STIX]{x2202}y=0$ . At this location, turbulence is maintained by the turbulent diffusion term. This transports the turbulence energy from the outer high energy region. The velocity–pressure gradient diffusion term transfers energy from the streamwise direction to the wall-normal and spanwise turbulence components. The wall-normal Reynolds stress $\langle v^{\prime }v^{\prime }\rangle$ has little production in the outer layer region (figure 20 c). The turbulence is mainly driven by the velocity–pressure gradient correlation term, which is balanced by the dissipation and turbulent diffusion terms. The spanwise Reynolds stress $\langle w^{\prime }w^{\prime }\rangle$ budget shows that the velocity–pressure gradient and convection terms are balanced with the dissipation and turbulent diffusion terms in the outer layer region (figure 20 d). The turbulent diffusion term transfers energy from the outer layer region to the inner layer maintaining turbulence around $y_{max}$ .

The shear stress budget $\langle u^{\prime }v^{\prime }\rangle$ has the production and velocity–pressure gradient as the dominant terms, which balance each other (figure 20 e). In the outer layer region, production is positive and is responsible for high values of shear stress $\langle u^{\prime }v^{\prime }\rangle$ . At $y=y_{max}$ the velocity gradient and production become zero and below this point production is negative, however shear stress remains positive for some distance in this region. The turbulent diffusion is responsible for a positive shear stress at and below $y=y_{max}$ , where production is zero or negative.

Figure 21. The inner-scaled velocity–pressure gradient correlation, pressure diffusion and pressure–strain correlation profiles in the near wall region for: (a) $tke=(\langle u^{\prime }u^{\prime }\rangle +\langle v^{\prime }v^{\prime }\rangle +\langle w^{\prime }w^{\prime }\rangle )/2$ ; (b) $\langle u^{\prime }u^{\prime }\rangle$ ; (c) $\langle v^{\prime }v^{\prime }\rangle$ ; (d) $\langle v^{\prime }v^{\prime }\rangle$ channel flow (Mansour, Kim & Moin Reference Mansour, Kim and Moin1988); (e) $\langle w^{\prime }w^{\prime }\rangle$ and (f) $\langle u^{\prime }v^{\prime }\rangle$ .

3.4.3 Velocity–pressure gradient and triple-velocity correlations

It has been shown from the budgets that for the wall jet, velocity–pressure gradient correlation and turbulent transport terms show the influence of the outer layer and depart from pure boundary layer behaviour. The velocity–pressure gradient correlation is responsible for energy redistribution among various Reynolds stresses. This term can be split as

(3.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{\langle u_{i}^{\prime }u_{j}^{\prime }\rangle }=-\left\langle \frac{\unicode[STIX]{x2202}p^{\prime }u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}p^{\prime }u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{j}}\right\rangle +\unicode[STIX]{x1D719}_{ij} & & \displaystyle\end{eqnarray}$$

and

(3.23) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{ij}=\left\langle p^{\prime }\left(\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{j}}\right)\right\rangle . & & \displaystyle\end{eqnarray}$$

The first term on the right-hand side in (3.22) is described as pressure diffusion or pressure transport and the second term $\unicode[STIX]{x1D719}_{ij}$ as pressure–strain correlation. The pressure strain term is the main redistributive component, which transfers energy from one component of turbulent kinetic energy to another. Negative values of pressure–strain correlation indicate loss and positive values indicate gain of energy by the corresponding Reynolds stress component. The trace of $\unicode[STIX]{x1D719}_{ij}$ is zero. Figure 21 shows the inner-scaled profiles for velocity–pressure gradient, pressure–strain and pressure diffusion term in the near wall region for turbulent kinetic energy, Reynolds normal and shear stresses. The terms $\unicode[STIX]{x1D6F9}_{\langle u_{i}^{\prime }u_{j}^{\prime }\rangle }$ , $\unicode[STIX]{x1D719}_{ij}$ and pressure diffusion for wall-normal Reynolds stress for a channel flow (Mansour et al. Reference Mansour, Kim and Moin1988) are also included (figure 21 d) for comparison. It is important to note here that in the current DNS velocity–pressure gradient and pressure–strain correlations are calculated explicitly and pressure diffusion is the difference of the two terms.

The pressure–strain for turbulent kinetic energy is zero (figure 21 a), which is the trace of $\unicode[STIX]{x1D719}_{ij}$ . It shows that the explicitly calculated pressure–strain is following the expected behaviour. In case of streamwise fluctuations the velocity–pressure gradient mainly consists of the pressure–strain correlation (figure 21 b), which is negative, indicating energy transfer from the streamwise to other directions.

The $\unicode[STIX]{x1D6F9}_{\langle u_{i}^{\prime }u_{j}^{\prime }\rangle }$ and $\unicode[STIX]{x1D719}_{ij}$ show more interesting behaviour for the Reynolds wall-normal and shear stresses. It has been shown for wall turbulence, e.g. in channel flow (figure 21 d) (Mansour et al. Reference Mansour, Kim and Moin1988), that for wall-normal Reynolds stress, velocity–pressure gradient and pressure–strain correlation follow the same trend and values, except very close to surface below $y^{+}=20$ . In the near wall region for $y^{+}<20$ the pressure–strain correlation changes sign and becomes negative. It transfers energy from the wall-normal direction to horizontal components of turbulence. The change of sign in pressure–strain correlation is associated with a process termed as ‘splat’ (Moin & Kim Reference Moin and Kim1982), where high negative vertical velocity comes close to the wall and creates a situation similar to jet impingement on a wall. On the other hand low speed vertical velocity moves turbulence away from the wall. However, negative vertical velocity brings more energy towards the wall than positive velocity takes away from the wall and excess energy transfers to the horizontal components. This wall-normal transfer of energy and splatting is associated with streak structures in wall bounded turbulence. As compared to channel flow, in the case of wall jets $\unicode[STIX]{x1D719}_{ij}$ is negative for $\langle v^{\prime }v^{\prime }\rangle$ across the whole inner layer. It has been shown from unsteady flow (figure 5) and various spectra (figure 18) that in the outer layer of wall jets there are large scale structures, which generate wall-normal impinging flow superimposed on the streak structures. A negative peak in $\unicode[STIX]{x1D719}_{ij}$ below $y^{+}<20$ is the outcome of this superposition. For spanwise $\langle w^{\prime }w^{\prime }\rangle$ , velocity–pressure gradient and pressure–strain correlation are identical and positive (figure 21 e). Since both $\unicode[STIX]{x1D719}_{11}$ and $\unicode[STIX]{x1D719}_{22}$ are negative, all the energy from these components transfers to the spanwise direction.

The $\unicode[STIX]{x1D6F9}_{\langle u_{i}^{\prime }u_{j}^{\prime }\rangle }$ , $\unicode[STIX]{x1D719}_{ij}$ and pressure diffusion terms for Reynolds shear stress $\langle u^{\prime }v^{\prime }\rangle$ transport are shown in figure 21(f). In case of channel flow wall turbulence (Moin & Kim Reference Moin and Kim1982) pressure–strain correlation has the opposite sign of Reynolds shear stress, except very close to the wall $(y^{+}<20)$ , where due to splatting, energy transfers to turbulence production. In the case of wall jets pressure–strain correlation has same sign as Reynolds stress in the entire inner layer region due to the interaction of outer layer.

Figure 22. Inner and outer-scaled profiles of triple-velocity correlation.

The profiles for the dominant terms of triple-velocity correlation $\langle u^{\prime }u^{\prime }u^{\prime }\rangle$ , $\langle u^{\prime }u^{\prime }v^{\prime }\rangle$ , $\langle v^{\prime }v^{\prime }v^{\prime }\rangle$ and $\langle u^{\prime }v^{\prime }v^{\prime }\rangle$ are shown in figure 22. These correlations appear in the turbulent diffusion term in the Reynolds stress budgets. The streamwise correlation $\langle u^{\prime }u^{\prime }u^{\prime }\rangle$ is the largest term but it has a weak influence on the turbulent diffusion, since its streamwise gradient is involved. The dominant term for the streamwise Reynolds stress budget is $\langle u^{\prime }u^{\prime }v^{\prime }\rangle$ . It is positive in the inner layer region below $y=y_{max}$ and negative in the outer layer region. It indicates that this term is responsible for transporting turbulence from high production regions of the inner and outer layers to the low production region around $y=y_{max}$ . The wall-normal velocity correlation $\langle v^{\prime }v^{\prime }v^{\prime }\rangle$ is negative below $y/y_{1/2}=0.8$ , which is responsible for turbulence transport from the outer layer to the inner layer. The Reynolds shear stress in the wall jet has a finite positive value at $y=y_{max}$ , where $\unicode[STIX]{x2202}\langle u\rangle /\unicode[STIX]{x2202}y=0$ and eddy viscosity models are not applicable. At this location, Reynolds shear stress is maintained by $\langle u^{\prime }v^{\prime }v^{\prime }\rangle$ , which is negative below $y/y_{1/2}=0.8$ and transfers positive shear stress from the outer layer to the inner layer.