Open Channel Flow Stratified by a Surface Heat Flux

OPEN CHANNEL FLOW STRATIFIED BY A SURFACE HEAT FLUX

John R. Taylor Department of Mechanical and Aerospace Engineering, University of California, San Diego La Jolla, CA [email protected]

Sutanu Sarkar Department of Mechanical and Aerospace Engineering, University of California, San Diego La Jolla, CA [email protected]

Vincenzo Armenio Dipartimento di Ingegneria Civile, Universita degli Studi di Trieste, Trieste, Italy [email protected]

ABSTRACT temperature difference ∆T across the channel. Nagaosa Large Eddy Simulation (LES) has been used to study flow and Saito (1997) considered a DNS of open channel flow driven by a constant pressure gradient in an open channel with fixed ∆T across the channel and a friction Reynolds with stable stratification imposed by a constant heat flux at number of 150. Komori et al. (1983) used steam to heat the free surface and an adiabatic bottom wall. Under these the surface of water in an inclined open channel, approxi- conditions a turbulent mixed layer develops underneath a mately equivalent to fixed temperature boundary conditions strongly stratified pycnocline. Turbulent properties in the (Garg et al. 2000). One of the common conclusions of pycnocline and at the free surface are examined as a function these studies is that for large ∆T applied across the channel, of the imposed stratification. It is found that increasing the near wall turbulent production is reduced by stratification. the friction Richardson number, a measure of the relative importance of stratification with respect to boundary layer Since all previous studies of stratified open channel flow turbulence, leads to a stronger, thicker pycnocline which have considered fixed temperature walls, the near-wall region eventually limits the impact of wall-generated turbulence on did not remain unstratified, and hence one of the major the free surface. Increasing stratification also leads to an influences of stratification was a reduction of the near-wall increase in the pressure-driven mean streamwise velocity near turbulence production. When considering environmental the free surface and a corresponding increase in mean shear, flows, the results of these studies may be analogous to the a decrease in the turbulent Reynolds stress, and a substantial atmospheric surface boundary layer under conditions of decrease in the eddy viscosity. PDF’s and visualizations strong surface cooling where a stably stratifying heat flux are employed to understand how stratification affects the at the ground can lower turbulent production in the surface interaction of bottom turbulence with the free surface. layer (Mahrt 1999). In contrast, our proposed boundary conditions are more relevant to the oceanic bottom boundary layer where the bounding surface is adiabatic, for example, see Lien and Sanford (2004) for a good explanation of the MOTIVATION differences between atmospheric and oceanic boundary layers. The present study considers open channel flow with stable stratification imposed by a constant heat flux at the free surface and an adiabatic lower wall. This choice of bound- FORMULATION ary conditions allows us to distinguish between buoyancy effects at the turbulence generation site and in the outer The geometry of the open channel considered here is shown region of the flow. Specifically, since the near wall region in Figure 1. Flow is driven by a uniform pressure gradient remains unstratified, the interaction between wall-generated aligned with the x-axis, and periodicity is applied in both turbulence and an external stable stratification is examined. horizontal directions while flat no-slip and no-stress surfaces Several previous studies have considered stratified channel bound the bottom and top respectively. For simplicity, the free flow, but in each case stratification was applied with fixed surface is assumed to be undeformed, a good approximation temperature boundaries. Armenio and Sarkar (2002) used for the low Froude number flow considered here. The y-axis a LES to study stratified closed channel flow with a fixed is aligned with the cross-stream direction, and the z-axis is normal to the wall. Velocities in the x,y, and z directions are denoted by u,v, and w and the domain size in the x and y direc- Table 1: Bulk Properties tions is 2πh and πh respectively, where h is the channel depth. Ri Ri Re Re P r The constant, negative density gradient imposed at the free τ b τ b surface can be thought of as surface heating with a constant 0 0 7132 heat flux if density changes are linearly related to temperature 25 0.04818 7150 ∗ 100 0.1927 7179 changes. The total density is given by ρT = ρ0 +ρ (x, t), with 400 5 ∗ 250 0.4818 7269 ρ << ρ0, allowing the Boussinesq approximation shown to be good for stratified water flows. 400 0.7708 7445 500 0.9635 7539

free surface, corresponding to stable stratiﬁcation. Gravity Constant Heat Flux At statistical steady state, the nondimensional density is a Horizontally Free Surface Periodic linear function of time, and can be decomposed:

z ∗ h ρ = ρ1(t) + ρ(x, t), (7) y x ph where ρ(x, t) is the turbulent density field that is statistically steady and ρ (t) denotes the deterministic field that decreases Adiabatic, No-Slip Wall 1 in time owing to the imposed surface heating. It can be shown 2ph that: t ρ1(t) = − . (8) Reτ P r Figure 1: Model Domain After each time integration of (1) - (3), the density change owing to ρ (t) is subtracted, and henceforth we will present re- The governing equations are nondimensionalized with the 1 1 sults concerning ρ(x, t), the statistically steady turbulent field. channel height h, friction velocity uτ = (τw/ρ0) 2 , and the absolute value of the imposed free surface gradient |∂ρ∗/∂z| . s The large eddy simulation (LES) used here is the same The shear stress, τ , used to define the friction velocity is the w as that used by Armenio and Sarkar (2002). A dynamic horizontally averaged value at the wall which must balance mixed subgrid model is used which has been shown to achieve the vertically integrated pressure gradient for steady state, proper dependence of the turbulent Prandtl number on the Πh =< τ >. With these choices, the nondimensional gov- w gradient Richardson number. The filtered equations are erning equations can be written: integrated using a version of the fractional-step method of 2 Zang et al. (1994), which is second order accurate in space Du ∗ ∇ u ∗ ˆ ˆ = −∇p + − Riτ ρ k + Πi, (1) and time. A dynamic eddy diffusivity model is used for the Dt Reτ subgrid density flux, see Armenio and Sarkar (2002) for more Dρ∗ ∇2ρ∗ details. = , (2) Dt Reτ P r ∇ · u = 0, (3) RESULTS dρ∗ z = 0 : u = v = w = 0, = 0 (4) dz ∂u ∂v dρ∗ Mean Profiles z = 1 : = = w = 0, = −1, (5) ∂z ∂z dz We begin by describing some mean flow properties. Aver- ages over the horizontal plane and time are denoted by < · >. where Π is the imposed pressure gradient equal to unity with The average streamwise velocity profile, nondimensionalized the present nondimensionalization, and the hydrostatic pres- by u , is shown versus z/h in Figure 2(a). Note that sure has been cancelled in the usual way. The nondimensional τ and the mean shear increase in a region near the free surface Reynolds, Richardson, and Prandtl numbers are defined as: where stratification is large. This is also reflected in the bulk u h g ∂ρ∗ h2 ν Reynolds number, Reb = Ubh/ν, shown in Table 1 to increase Re = τ , Ri = − , P r = , (6) τ τ 2 with bulk Richardson number. Nagaosa and Saito (1997) also ν ρ0 ∂z s uτ κ observe an increase in the streamwise velocity when they apply where κ is the molecular diffusivity. Notice that since the a fixed temperature difference across the channel to produce imposed surface density gradient is used to make the density stable stratification. The region of increased velocity in their nondimensional, it appears in the Richardson number defined case extends from the surface to about 10 wall units from the in (6); therefore increasing Riτ is physically equivalent to lower wall, a much thicker region than is seen here. A conve- increasing the imposed surface stratification. When Riτ = 0, nient measure of the bulk change in streamwise velocity is the density acts as a passive scalar and the velocity field can be skin-friction coefficient: checked against previous unstratified open channel studies. 2 When Riτ > 0, a negative density gradient is imposed at the Cf = 2τw/ρUb . (9) Table 2 gives C for each case of Ri . For comparison, the val- (a) (b) f τ 1 25 ues found by Nagaosa and Saito (Nagaosa and Saito 1997) are 0.9 also shown. Riτ,∆ defined with the density difference across the channel, 0.8 Rit = 0 20 gh∆ρ Rit = 100 Ri = , (10) 0.7 τ,∆ 2 Rit = 250 ρ0uτ Rit = 400 0.6 t 15 is introduced to measure stratification on a similar basis in Rit = 500 U / z/h 0.5 all studies. Clearly Cf decreases with Riτ,∆ in both studies, but the dependence observed here is much weaker than the 0.4 10 31% decrease between Riτ,∆ = 0 and 20 observed by Nagaosa 0.3 and Saito (1997). This can be explained by the relatively 0.2 5 limited region affected by stratification in the present study, a qualitative difference with respect to the previous fixed ∆T 0.1 cases. 0 0 0 5 10 15 20 25 100 101 102 103 + U/Ut z Table 2: Skin-friction coefficient Figure 2: Mean Velocity Profile Taylor et al., 2005 Nagaosa and Saito, 1997 (a) (b) 1 3 3 Riτ Riτ,∆ Cf ∗ 10 Riτ,∆ Cf ∗ 10 0.9 0.2 0 0 6.291 0 8.71 0.18 25 0.67 6.258 10 7.06 0.8 0.16 100 3.1 6.213 20 6.03 0.7

r 250 12.9 6.054 0.14 0.6

400 33 5.765 lamina 0.12

z/h 0.5 Ri = 0 500 51.3 5.629 t 0.1 Ri = 100 0.4 t Dr/Dr 0.08 Rit= 250 0.3 Rit= 400 0.06 Ri = 500 0.2 t 0.04 Laminar The averaged density profile for each case is plotted as a 0.1 Solution 0.02 function of nondimensional height in Figure 3(a) where the 0 0 density is made nondimensional by ∆ρ, the difference between -1 -0.8 -0.6 -0.4 -0.2 0 0 100 200 300 400 500 r r Dr wall and surface values as in Komori (1983). The laminar ( - z=0)/ Rit solution to the density equation with the appropriate boundary conditions is also shown. Unlike the gradual variation of Figure 3: Mean density profiles and density difference across ρ(z) in the laminar case, the turbulent flow exhibits a strongly channel stratified region, or pycnocline, near the free surface that over- the stability of the flow, with linear instability possible only lies a well-mixed region near the lower wall. The presence of if Rig < 1/4 somewhere in the domain. Knowing that the the well-mixed region must depend on the existence of active stratification near the surface changes significantly with Riτ , turbulence since the density gradient of the laminar solution it is surprising that above the linear stability threshold, vanishes only near the wall. The thickness of the pycnocline in- Rig is nearly independent of Riτ . Evidently the increase creases with Riτ , implying that the turbulence generated near in mean shear compensates for the increase in N in this region. the lower wall is less effective at mixing for large Riτ . Figure 3(b) shows the variation of ∆ρ between cases. ∆ρ tends to increase with increasing Riτ (increasing stabilization) but, even for the largest Riτ = 500 considered here, ∆ρ is much smaller TURBULENCE CHARACTERISTICS than in the laminar case. Figure 5(a) shows the profile of the rms vertical velocity. In the lower half of the channel, the profiles collapse and The buoyancy or Brunt-Vaisala frequency, N defined as: are consistent with unstratified closed channel flow. In the upper region, wrms decreases monotonically with increasing 2 −g ∂ < ρ > N = , (11) Riτ . Since wrms corresponds to the vertical turbulent ρ0 ∂z kinetic energy, and Riτ is linked to the size of the buoyancy is shown in the left panel of figure 4. This plot makes clear suppression term in the TKE budget, the observed decrease the deepening and strengthening of the pycnocline with is as anticipated. Interestingly, near the free surface where increasing Riτ . A local measure of the relative importance of wrms is supressed by the geometry, the dependence on Riτ is stratification and shear, the gradient Richardson number can lost. be defined using N and the mean shear, S = d /dz so 2 2 0 0 that Rig = N /S , is also shown in figure 4. The gradient The Reynolds shear stress, is shown in Richardson number measures the relative importance of Figure 5(b). For comparison, the total shear stress, turbulent production by the mean shear, and suppression τ(z) = τwall(1 − z/h) is also plotted. It can be shown (eg. by the stable stratification. As such, it is associated with Pope (2000)) that the viscous shear stress is the difference 1 1 Rit= 0 Rit= 500 0.9 0.9 - 0.11 0.022 - 0.089 0.026 0.8 0.8 1 1 0.7 0.7 t t 0 0 0.6 0.6 w ’ / u w ’ / u Rit = 25 z / h 0.5 Rit = 100 z / h 0.5 -1 -1 Ri = 250 0.4 t 0.4 0.028 - 0.064 0.03 - 0.052 Rit = 400 4 2 0 2 4 4 2 0 2 4 0.3 Ri = 500 0.3 t u’/ut u’/ut 0.2 0.2 0.1 0.1 Figure 6: u’ vs w’ at z/h=0.84 0 0 . 0 10 20 30 -1 0 2 10 10 10 The mean streamwise stress balance can then be written: N Rig z d 1 τw(1 − ) = ( + νT ), (13) Figure 4: Brunt-Vaisala Frequency and Gradient Richardson h dz Reτ number so any change in the mean shear between cases must also be between this line and < u0w0 >. Thus, the increase in the reflected in the eddy viscosity, plotted in Figure 7(a). Eddy mean vertical shear (equivalently viscous shear stress) in viscosity decreases very significantly with Riτ , even in the in- the pycnocline, and therefore at the free surface, terior of the open channel where stratification is relatively low. occurs because of the stratification induced decrease in the 0 0 magnitude of , which is strongest when Riτ = 500. Figure 7 shows the buoyancy flux, < ρ0w0 > nondimension- The drop in < u0w0 > magnitude will be explained using alized by the free surface density gradient, the channel height, energy arguments in the section on turbulence-surface and uτ . Vertical motion under the negative mean density interactions. gradient implies a positive buoyancy flux for the usual case of co-gradient transport. The buoyancy flux decreases every- (a) (b) where with increasing Riτ and has a small countergradient 1 1 value near the surface when Riτ = 500. Countergradient 0.9 0.9 transport is associated with falling heavy fluid that releases 0.8 0.8 potential energy to kinetic energy. Komori et al. (1983) also find a countergradient heat flux, although they report it being 0.7 0.7 much larger and appearing at lower Riτ than in the present 0.6 0.6 simulations. The difference is presumably due to the boundary conditions, since in the Komori et al. (1983) experiments, z / h z / h 0.5 Ri = 0 0.5 t the wall and free surface were roughly held at fixed tempera- 0.4 Rit= 100 0.4 Ri = 250 ture. Large countergradient buoyancy fluxes were also seen in 0.3 t 0.3 Rit= 500 the study by Armenio and Sarkar (2002) in a closed channel 0.2 0.2 with fixed temperature boundary conditions at the walls. The 0.1 0.1 mass diffusivity, κT defined as:

0 0 0 0 dρ 0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.5 0.0 < w ρ >= −κT , (14) 2 dz wrms / ut / ut also decreases very significantly with Riτ at nearly every Figure 5: rms vertical velocity and Reynolds shear stress vertical level (not shown). Contributions to the Reynolds stress can be seen by plotting u0 vs. w0 as shown in Figure 6 for z/h = 0.84. In each quadrant of the plots is a label showing its con- TURBULENCE-SURFACE INTERACTIONS 0 0 2 tribution to /uτ . The upwelling events can be The increase in seen near the free surface in clearly seen for Riτ = 0 as an anisotropic tail extending the highly stratified cases can be attributed to a potential to the upper left. When Riτ = 500 the strength of the energy barrier owing to the presence of the pycnocline. It upwellings is diminished, and the distribution becomes has been shown previously (Pan and Banerjee 1995) that a more isotropic. In both cases, downwelling events are not as large portion of the Reynolds stress near an unstratified free energetic as upwelling bursts, and contribute less to < u0w0 >. surface in open channel flow is due to impinging of low-speed fluid advected from the near wall region. While the wall While it has been seen that the influence of Riτ on Cf is generated low speed streaks do not maintain coherence over rather small, the local turbulent diffusion is strongly affected distances comparable to the channel height in this study, in a significant portion of the channel, as can be seen by con- low-speed ejections from the wall boundary layer are observed sidering the eddy viscosity, νT : to directly impact the free surface in the low stratification cases. 0 0 d − = νT . (12) dz (a) (b) 1 1 1 0.9 0.9 0.9 0.8 0.8 0.7 0.7 0.8 0.6 0.6 0.7

0.5 z / h 0.5 0.6 0.4 0.4

Ri = 0 z 0.3 0.3 t 0.5 Rit= 100 0.2 0.2 Rit= 250 0.4 Ri = 400 0.1 0.1 t Rit= 500 0.3 0 0 -4 0 20 40 60 0 2 4 x10 0.2 n n T/( ) /|dr/dz|suth 0.1 Figure 7: Eddy viscosity and buoyancy flux 0 -2 -1 0 1 That the upward advection of low speed fluid to the 10 10 10 10 surface is inhibited for large Riτ is implied by the drop in PE / Vertical TKE correlation between u0 and w0 in the Reynolds stress of figure 5(b). To determine the fate of turbulence generated near the Figure 8: Ratio of potential energy deficit to vertical TKE lower wall more directly, it is useful to consider an energy balance. Traditionally, the buoyancy scale wrms/N gives a The strength and frequency of upwelling events can measure of how far a fluid parcel would travel vertically if be quantified with a joint probability density function all of its vertical turbulent kinetic energy were converted to (PDF) of the vertical velocity, w, and density anomaly, 0 potential energy. For the situation considered here, this is not ρ (x, t) = ρ(x, t)− < ρ > (z) as shown in Figure 9 for Riτ = 0 accurate since N is highly variable in the vertical direction. and 500 at z/h = 0.975. The figure caption lists the values of For instance, in the highly active region near the lower wall, ρ0 corresponding to the mean density at the top and bottom wrms is large while N is small, so the buoyancy scale may for comparison. The plot indicates that when Riτ = 500, it be very large. However, the presence of a strong pycnocline is very rare for fluid with density equal to the mean at z = 0 near the surface adds to the potential energy barrier, and (corresponding to ρ0 = 0.077 and well out of the plotted may prevent direct interaction with the surface. region) to be seen at this height. For the case of Riτ = 0 it is common to see ρ0 = 0.008, the mean at z = 0, and the free As a more accurate measure of the ability of local tur- surface value ρ0 = −0.014 is somewhat less likely. The tails 0 bulence to reach the free surface, we compare the vertical of the w distribution are wider when Riτ = 0 and ρ > 0; a turbulent kinetic energy (TKE) to the potential energy deficit large w and ρ0 > 0.008 is associated with the strong upwelling relative to the free surface. This ratio, plotted in the left panel events seen when Riτ = 0 and mentioned previously. For the of figure 8 is: case of Riτ = 0 large w events of both signs are associated with positive density anomaly and the distribution is nearly g R h(< ρ > (z)− < ρ > (z0))dz0 symmetric about w = 0. Evidently, at this location, the z . (15) 1 0 0 downwellings of dense fluid are as strong and frequent as the 2 < w w > upwellings. When Riτ = 500, the largest vertical velocities

As expected, this ratio is largest when Riτ = 500 since are no more likely to be associated preferentially with either this case has a stronger, deeper pycnocline, requiring more heavy or light fluid, indicating that events with upwelling of energy to reach the free surface. The cases with the lowest dense fluid are not dominant. The off-centered maxima of the 0 0 stratification, namely Riτ = 25 and Riτ = 100, have small ρ distributions are presumably due to the asymmetry in ρ as- values of this ratio. That the pycnocline is weak in relation sociated with the top and bottom of the channel for each case. to the vertical TKE helps to explain why these cases are quite similar to the passive scalar case, Riτ = 0. The case The effect of stratification on dense fluid upwellings near of Riτ = 250 appears to be ‘transitional’ since the energy the free surface can also be clearly seen by examining the ratio is about one at most locations. Meanwhile, the strength instantaneous property distributions. Figure 10 shows ρ0 and of the pycnocline dominates over the vertical TKE when w0, the deviation from the horizontal mean, at z/h = 0.999 Riτ = 500. Since the low-speed fluid near the wall, on the for Riτ = 0 and Riτ = 500 at the last simulation time average, does not have sufficient energy to reach the surface in both cases. The height of the surface mesh denotes in the latter case, a drop in < u0w0 > is observed near the the vertical velocity with the tall peaks indicating rising surface and, correspondingly, there is an increase in . It fluid (w0 > 0). The corresponding grayscale shows ρ0 with should be noted that since this ratio is an average measure, dark gray denoting heavy fluid with positive ρ0. Notice it does not preclude the instantaneous advection of bottom that for Riτ = 0, each region of upwelling is associated fluid to the surface, but does indicate that it is much less with a positive density anomaly indicating an upwelling of likely when a strong pycnocline exists. dense fluid from the bottom. When Riτ = 500 none of the positive w0 patches at this particular time are associated effects of changing the friction Richardson number, Riτ , are (a) Rit=0 (b) Rit=500 examined. In all cases, a stably stratified pycnocline overlies a lower region that is well mixed by turbulence generated

10 10 0.015 0.015 20 at the lower wall. As Riτ is increased, the turbulence in 20 10 10 0.01 30 40 0.01 20 the mixed region remains unchanged while the turbulence in 30 40 50 10 the pycnocline is aﬀected by buoyancy, but never completely 30 20 s

s 0.005 40 0.005

60 30

/ d z | h 10 / d z | h 0 suppressed. It is possible that by suﬃciently increasing Riτ , 2

50 r

50 r 0 0 10 10 20

50 the ﬂow in the pycnocline could relaminarize, although this

40 40 ’ / | d ’ / | d 60 20

r 40 r -0.005 -0.005 limit is not obtained here. It is observed that increasing Riτ 20 30 -0.01 30 10 -0.01 results in an increase in the bulk Reynolds number, Reb, 20 10 -0.015 -0.015 10 and a deepening and strengthening of the pycnocline. The 20 mean velocity deviates from the log law with the extent of -0.04 -0.02 0 0.02 0.04 -0.04-0.02 0 0.02 0.04 the deviation systematically increasing with Riτ . Since the w w/ut /ut gradient Richardson number is too large in the pycnocline for local turbulent production, the influence of increasing Figure 9: Joint PDF between vertical velocity, w and density Riτ can be explained by a potential energy barrier affecting anomaly, ρ0(x, t) = ρ(x, t)− < ρ > (z) at z/h=0.975 for (a) the interaction of bottom boundary layer turbulence with 0 Riτ = 0, (b) Riτ = 500. The density anomalies corresponding the surface region. Visualizations and joint PDFs of ρ and to < ρ > at the top and bottom respectively are -0.014 and w show that upwelling of dense bottom fluid to the surface 0.008 for Riτ = 0, and -0.026 and 0.077 for Riτ = 500. becomes rare in the large Riτ cases. with large positive ρ0. These snapshots are typical of those seen throughout the simulation; while the existence of dense fluid upwellings cannot be precluded for the strongest strati- * fication cases, they are much less common than when Riτ = 0. References

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0.5 Pope, S. (2000). Turbulent Flows. Cambridge: Cambridge 6 University Press. 0 3 4 5 0 1 2 x /h Zang, Y., R. Street, and R. Koseﬀ (1994). A non-staggered grid, fractional step method for time-dependent incom- Figure 10: Instantaneous height maps of vertical velocity with pressible Navier-Stokes equations in curvilinear coordi- density perturbation in grayscale nates. J. Comput. Physics 114, 18–33.

CONCLUSION Turbulent open channel ﬂow with an imposed density gradient at the free surface corresponding to surface heating and an adiabatic bottom boundary is studied here and the