Velocity-Vorticity Decomposition of the Reynolds Stress Gradient in a Shear Layer Scott C. Morris Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, IN 46556 USA [email protected] John F. Foss, Aren Hellum Department of Mechanical Engineering Michigan State University East Lansing, MI 48824 USA [email protected] ABSTRACT The single stream shear layer provides an interesting Time series measurements of velocity and vorticity using a boundary value problem in which to study the Reynolds four sensor hot-wire technique have been acquired in a large stresses. The only non-zero component of the Reynolds shear scale single stream shear layer. The Reynolds stress gradient stress tensor is u'v' . This term appears in the boundary layer which appears in the time averaged momentum equation can form of the momentum equation: be written in terms of velocity-vorticity products. The experi- mental data allows for the evaluation of these quantities, and ∂u ∂u ∂u'v' u------ +-v------ = ----------- (2) the spectral content of the time series provides insight into the ∂x ∂y ∂y scales of motion which lead to the time averaged values. which, for the flow field considered herein, appropriately The power spectral density function for the vorticity time assumes that the mean viscous stress, pressure gradients, and series (ω , ω ) show substantial values at a wide range of wave z y streamwise gradients are negligible. The present examination numbers. The velocity-vorticity products also indicates a wide of the Reynolds stress gradient begins with the identity (again, distribution in wave number space. This indicates the exist- ignoring streamwise gradients): ence of anisotropic motions which contribute to the Reynolds shear stress gradient at wave numbers at both very high and ∂u'v' ------------ ≈ v'ω' -w'ω' .(3) very low wave numbers. ∂y z y Tennekes and Lumley [2] provide an interpretation of this INTRODUCTION expression in terms of a pseudo body force which results from The Reynolds averaged Navier-Stokes equations can be the velocity-vorticity correlations. Equation (3) then indicates written as: that the net shear stress gradient is a result of contributions from (or cancellations of) the two velocity-vorticity products. ∂ui 1 ∂ The subject of this paper is the direct measurement of the u -------- = ---------- ()-Pδ + 2µS -ρu' u' (1) time series data of the terms of equation 3. This provides a j∂x ρ∂x ij ij i j j j unique opportunity to observe the scales of motion (in terms of spectral content) of the gradient of the Reynolds stress. where Sij represents the time averaged rate of strain tensor. The modeling of the Reynolds stress u'i u'j is the subject of DESCRIPTION OF EXPERIMENT considerable research; see, e.g., Pope 2000 for a review. The A large scale single stream shear layer facility located at physical mechanisms which lead to the Reynolds stresses are the Turbulent Shear Flows Laboratory of Michigan State Uni- still a topic of research, since no universal model has been versity was used for the experiments; see Figure 1. The Rey- found that relates these terms to the mean flow variables. nolds number of the boundary layer at separation is Reθ=4650, Empirical and heuristic models often encounter problems based on the momentum thickness θ(x=0)=θo=9.6mm, and the when varied physical mechanisms are responsible for the gen- constant free stream velocity: Uo=7.1m/s. The shear layer test eration of Reynolds stresses within a given domain. section is 9.7m in length from the separation point to the tun- nel exit. A low disturbance level entrainment flow is delivered through four axial fans to maintain a zero pressure gradient test section. Spanwise vorticity measurements have been acquired using the compact four sensor hot-wire probe developed by Foss and coworkers, see, e.g., Haw et al. (1989) and demon- strated by Wallace and Foss (1995). A schematic representa- tion of the probe is shown in Figure 2. The two parallel wires record the magnitude of velocity with a small (δy~1.4mm) spatial separation. The sensors configured as an X-array are used to recover the v component of velocity. From these three signals, a micro circulation domain is constructed with the flow direction calculated using a convected distance from the local (in time) velocity magnitude. This is shown schemati- cally in Figure 3. The resolution of the micro circulation Figure 2. Schematic of the four wire vorticity probe. Note the domain in the present experiment is approximately 7 Kolmog- distance between the parallel wires is typically 1.5mm. orov length scales. The vorticity probe was traversed across the shear layer at the streamwise location: x/θo=675. Time series data were acquired at 40kHz for 50 seconds. The Reynolds number of the shear layer at this location was Reθ=150,000 based on the local momentum thickness. Primary Inflow Flow Conditioners Contraction Gap for contraction boundary layer removal Uo Figure 3. Diagram of the micro circulation domain used to calculate vorticity from the four wire probe. 5.74m Boundary Layer Plate Separation Edge RESULTS AND DISCUSSION The time averaged velocity distribution: uy(), exhibits Flow Conditioners excellent self-preservation (x/θ(0)>100), see Fig. 4. The mean velocity field, with the assumption of self similarity, can be y used to evaluate the non-dimensional Reynolds stress: x 2 Entrainment hy()≡ ()1 ⁄ σ []u'v' ⁄ Uo shown in Fig. 5 along with the Fans directly measured values. The Reynods stress gradient: 2 h'()y ≡ ()∂1 ⁄ σ []∂u'v' ⁄ Uo ⁄ ()y ⁄ θ , where σ=dθ/dx and η=(y-y1/2)/θ is shown in Fig. 6. The measured values obtained at x/θo=675, η=0 will be 9.75m used for the present communication. Several statistics of inter- est at this location include, where [~] indicates standard devia- Bypass holes tion: ˜ ˜ uU⁄ o = 0.508 , u ⁄ Uo = 0.157 , v ⁄ Uo = 0.118 ˜ ˜ ˜ Uo w ⁄ Uo = 0.136 , u'v' ⁄ ()uv = 0.413 ˜ ˜ ˜ ˜ ωzλu ⁄ u = 0.92 , ωyλu ⁄ u = 0.58 Tunnel Exit where λ is the Taylor microscale and the vorticity scalings Figure 1. Schematic of the single stream shear layer facility. u reflect the dissipative scales of the flow field. A more sensitive (pre-multiplied by k1) representation of 1.2 the component spectra k1Eγγ(k1η) is presented in Fig 6 for γ= 1, 2, 3. Note that when spectra are plotted in this form, equal u 1.0 areas “under the curve” contribute equally to the signal vari- x/θ =370 U o ance. As expected the E11 values exhibit the largest magni- o0.8 x/θo=650 tudes at low wave number. Fig. 7 presents a segment of the complete time record 0.6 acquired for the ωz(t) magnitudes at η=0.508. This quantity can be described using the Reynolds decomposition: 0.4 ωz()t = ωz + ω'z()t . Quite unlike the velocity distributions ˜ 0.2 the ωz ⁄ ωz are quite large (≈22 at this location). It is instruc- tive to consider not only the large fluctuation levels but also 0.0 -4 -3 -2 -1 0 1 2 3 4 the rapidly changing values. It is implied that the instanta- η neous vorticity field is highly stretched by the velocity field []V ⋅ ∇ω . Figure 4. Mean velocity profile in the shear layer Fig. 8 presents the pre-multiplied vorticity spectral distri- 0.25 butions. Since the area under the curve in this representation is the normalized mean square vorticity value, the evident differ- -h(η) x/θ =384 o ences in the indicated area reflects the difference in the nor- 0.20 x/θo=675 malized standard deviations for ωz and ωy 0.92 cf 0.58, respectively. The other noteworthy features of Fig. 7 are: 0.15 i) quite similar magnitudes for k1η>0.2, ii)a peaked ωy spectra (at k1η=0.15) is evident whereas the 0.10 ωz spectra is “nominally invariant” over a wide range of k1 values. 0.05 The velocity-vorticity co-spectral distributions that directly address the Reynolds stress gradients are shown in Fig. 9. The v, ωz cospectra clearly shows larger values in the 0.00 -3 -2 -1 0 1 2 3 4 low k1 (k1η=0.001) region. These values are essentially 2 η responsible for the larger v'ωz' value (221 m/sec or Figure 5. Calculated and measured Reynolds shear stress 2 θv'ωz' ⁄ Uo =0.199) that stands in contrast to the w'ωy' value 0.100 2 2 (141 m/sec or θw'ωy' ⁄ Uo=0.087). The strong similarity of 0.050 the pre-multiplied E22, E33 spectra, in contrast with the differ- ent low frequency content of the vorticity-velocity spectra, -h'(η) 0.000 clearly identify the mechanistic contributors to the Reynolds stress gradients. Namely, the motions whose k1η values are of -0.050 order 10-3 primarily contribute to the Reynolds stress gradient. -0.100 The scaling used above: [θ and Uo], result in a non-dimen- sional difference for the rhs of Equation 3 as: -0.150 θ -3 -2 -1 0 1 2 3 4 ------ []v'ω ' – w'ω ' ==0.199– 0.087 0.112 . 2 z y U η o This magnitude can be compared with the measured gradient: Figure 6. Reynolds stress gradient, obtained by fitting and ∂ u'v' differentiating the data shown in Fig. 5. ----------------- -------- = 0.0021 ∂()y ⁄ θ 2 Uo The wave number content of the present measurements will be used to characterize the physical attributes of this flow The implication of these data is quite apparent.