Experimental investigation on destruction of Reynolds stress in a plane jet

Abstract

Flow structures that cause destruction of the Reynolds (shear) stress are investigated by means of simultaneous measurement of the velocity gradient and pressure. Two types of combined probes are used for the simultaneous measurement of the velocity gradient and pressure. One consists of an X-type hot-wire probe and a pressure probe; another consists of two I-type hot-wire probes arranged in parallel with a small vertical separation (double-I) and a pressure probe. The hot-wire probe and the pressure probe are aligned along the streamwise direction for both combined probes. We confirm that the simultaneous measurement of the velocity gradient and pressure in a plane jet is satisfactorily performed using the combined probes, except for the outside of the velocity half-width. In a plane jet, $\overline{(p'/\rho )(\partial u'_{2}/\partial x_{1})}$ does not contribute to the Reynolds stress destruction, so that the total pressure-rate-of-strain correlation is almost equal to $\overline{(p'/\rho )(\partial u'_{1}/\partial x_{2})}$. The flow structures that destruct the Reynolds stress are extracted by the conditional sampling technique. We find two structures, which support the thought experiment conducted by Hinze (Turbulence, 2nd edn, McGraw-Hill College, Pennsylvania, 1975). The structures are concentrated around large-scale vortex structures, which cause strong momentum transfer. The scale of the structures is intermediate: smaller than large-scale motion contributing to the momentum transfer, and larger than small-scale motion contributing to the energy dissipation.

Graphical abstract

Evaluation of the error due to the disagreement of the measurement points of velocity gradient and pressure

Fig. 19

Cospectrum of $p'/\rho$ and $\partial u_{2}'/\partial x_{1}$ with and without correction for the disagreement of the measurement points

For the present combined probes, the measurement point of the velocity gradient is 2.9 mm upstream of that of the pressure. In this study, the disagreement of the measurement points is corrected by shifting the measured signal in the streamwise direction after the time–space conversion. This appendix evaluates the error caused by measuring the velocity gradient and the pressure at the different points.

There are two requirements to correct the disagreement of the measurement points by time–space conversion (regardless of TH base or MTH base) and by shifting the measured signal:

• 1. The turbulent structure is frozen during traveling from the velocity measurement point to the pressure measurement point.

• 2. The turbulent structure is convected only in the streamwise direction.

(1) It may not be satisfied by small-scale structures, because their timescales are so small that the structures can be easily deformed during the traveling. (2) It is not realized due to the three-dimensional fluctuation of the convection velocity, especially when the relative turbulence intensity is large just like the present plane jet.

First, let us estimate the scale at which the structures can be regarded to be frozen. Here, we consider to correct the result at $x_{2}/b_{u}=0.8$ using the time–space conversion based on the TH in Eq. (1); that is, we use the unique convection velocity $U_{co}=U_{1}=3.2$ m/s. A structure measured by the hot-wire probe reaches the pressure measurement point after “traveling time", $2.9\times 10^{-3}/3.2=9.0\times 10^{-4}$ s. The timescale of the structure must be sufficiently larger than the traveling time, so that the structure is frozen during the traveling.

The timescale $\tau$ of the turbulent structure at $k_{1}$ is estimated to be (Tennekes and Lumley 1972, p. 260):

$$\begin{aligned} \tau \sim {k_{1}}^{-3/2}{E_{11}(k_{1})}^{-1/2}. \end{aligned}$$

(7)

Kolmogorov scaling $E_{11}\propto \varepsilon ^{2/3}k_{1}^{-5/3}$ gives $\tau \sim \varepsilon ^{-1/3}k_{1}^{-2/3}$.

Here, we evaluate two timescale at $k_{1}\eta =1$ (Kolmogorov timescale) and $k_{1}\eta =0.1$. The Kolmogorov timescale is specifically $8.63\times 10^{-4}$ s, which is almost the same as the traveling time. Therefore, the smallest turbulent structures cannot be measured even if the present correction is performed, because they can be deformed during the traveling. On the other hand, the specific timescale for $k_{1}\eta =0.1$ is $\tau =4.5\times 10^{-3}$ s, which is considerably larger than the traveling time. This indicates that, at least, the structures in $k_{1}\eta \le 0.1$ are almost frozen during the traveling from the X probe to the pressure probe.

Second, we discuss the influence of fluctuation of the convection velocity. The fluctuation of the convection velocity in the streamwise direction can be corrected by use of the MTH (Kahalerras et al. 1998; Burattini et al. 2005; Djenidi et al. 2016), but the fluctuation in the cross-streamwise and spanwise directions cannot. However, the error caused by the fluctuation in the vertical direction is expected to be no more than the error by the fluctuation in the streamwise direction because of $u'_{1,\text {rms}}>u'_{2,\text {rms}}$ in the plane jet.

The influence the fluctuation of the convection velocity is checked by comparison of cospectra $C_{p/\rho ,\partial u_{2}/\partial x_{1}}(k_{1})$: one is corrected based on TH (Eq. 1) and the other is based on MTH (Eq. 2). The discrepancy between the two cospectra is the influence of the fluctuation of the convection velocity in the streamwise direction. Figure 19 shows $C_{p/\rho ,\partial u_{2}/\partial x_{1}}(k_{1})$ at $x_{1}/h=40$, $x_{2}/b_{u}=0.8$. The $C_{p/\rho ,\partial u_{2}/\partial x_{1}}(k_{1})$ without any correction is also included in Fig. 19 for reference. The cospectrum corrected by the MTH is almost the same as that by TH. We can confirm that the error in measuring $\overline{(p'/\rho )(\partial u'_{1}/\partial x_{2})}$ due to the fluctuation of the convection velocity is actually small.