Turbulence and waves in numerically simulated slope flows

Transcription

1 9 ème Congrè Françai de Mécanique Mareille, 4-8 août 9 Turbulence and wave in numerically imulated lope flow E. FEDOROVICH, A. SHAPIRO School of Meteorology, Univertiy of Oklahoma, David L. Boren Blvd., NORMAN (USA) Abtract : Direct numerical imulation (DNS) i applied to invetigate propertie of katabatic and anabatic flow along thermally perturbed (in term of urface buoyancy flux) loping urface in the abence of rotation. Numerical experiment are conducted for homogeneou urface forcing over infinite planar lope. The imulated flow are the turbulent analog of the Prandtl (94) one-dimenional laminar lope flow. The imulated flow achieve quai-teady periodic regime at large time, with turbulent fluctuation being modified by peritent low-frequency ocillatory motion with frequency equal to the product of the ambient buoyancy frequency and the ine of the lope angle. Thee ocillatory wave-type motion reult from interaction between turbulence and ambient table tratification depite the temporal contancy of the urface buoyant forcing. The tructure of the mean-flow field and turbulence tatitic in imulated lope flow i analyzed. An integral dynamic imilarity contraint for teady lope/wall flow forced by urface buoyancy flux i propoed and quantitatively verified againt the DNS data. Key word : katabatic, anabatic, numerical imulation, boundary layer, turbulence Introduction Slope wind (flow) are typical for vat area of the Earth, and often play an important role in the weather and climate of thee area. From the tandpoint of baic fluid dynamic, lope wind are buoyantly driven boundary-layer-type flow along heated or cooled loping urface in a tratified fluid. It i common to ditinguih between anabatic wind, which are driven by urface heating, and katabatic wind, which reult from urface cooling. Slope flow conflate three characteritic apect of geophyical fluid dynamic: buoyant forcing, tratification, and turbulence. There are till many open quetion regarding the tructure and propertie of thee flow. Of particular interet for practical application are the mean flow and turbulence tructure of lope wind a function of urface thermal forcing and lope angle. An early miletone in the conceptual undertanding of lope flow wa the development of the Prandtl [] one-dimenional model for a flow of a vicou tably-tratified fluid along a uniformly cooled or heated loping planar urface. Flow in the model ha a boundary-layer character (low-level jet topped by weak revered flow). The along-lope advection of environmental (mean) temperature balance thermal diffuion, and the along-lope component of buoyancy balance diffuion of along-lope momentum. Obervation ugget that, with appropriately tuned parameter, thi imple model provide a good decription of the vertical tructure of katabatic flow and a reaonable approximation of anabatic flow. In the preent tudy, the tructural feature of turbulent anabatic and katabatic flow along infinite planar lope are invetigated by mean of direct numerical imulation (DNS). The DNS experiment explore prototypical phyical effect in idealized downcaled atmopheric lope flow. Particular emphai i placed on tudying the dependence of the mean profile and turbulence tatitic in lope flow on the ign and magnitude of urface buoyancy forcing (in term of urface buoyancy flux) and the lope teepne. Equation of lope-flow dynamic The momentum balance equation for a mall-cale (very large Roby number) flow in the Bouineq approximation are the following: u u u u π u u u + u + v + w = + binα + ν, () t x y z x x y z

2 9 ème Congrè Françai de Mécanique Mareille, 4-8 août 9 v v v v v v v u v w π = + ν + +, () t x y z y x y z w w w w w w w + u + v + w = π + bcoα + ν, (3) t x y z z x y z with the heat/buoyancy balance preented by b b b b b b b + u + v + w = N ( uinα + wco α) + ν h, (4) t x y z x y z and ma conervation repreented by the continuity equation for an incompreible fluid, u v w + + =. (5) x y z In the above equation, u, v, w are velocity component in the right-hand lope-following Carteian coordinate ytem x (uplope), y (cro-lope), z (lope-normal), π = [ p pe( z')]/ ρr i the normalized preure perturbation [ pe( z ') i the environmental preure, z ' i the true vertical coordinate, ρ r =cont i the reference denity value], b βθ i the buoyancy with θ = Θ Θ e ( z ') being the potential temperature perturbation and β = g / Θr the buoyancy parameter ( Θ r =cont i the reference potential temperature value, g i the gravity acceleration), γ = dθe / dz' =cont i the gradient of environmental potential temperature, α / i the lope angle, ν i the kinematic vicoity, ν h i the thermal diffuivity, and N = ( βγ ) i the Brunt- Väiälä (buoyancy) frequency. The lateral boundary condition for prognotic variable (u, v, w, b) and normalized preure π are periodic (the loping urface i uppoed to be doubly-infinite along x and y). The upper boundary condition (large z) are ϕ / z =, where ϕ / z = i any of (u, v, w, b), and π / z i obtained from (3). The urface (z=) condition are no-lip and impermeability (u=v=w=), with π / z obtained from (3), and ν h( b/ z) = B, where B i the urface buoyancy flux which alo ha a meaning of the urface energy production rate. 3 Numerical imulation of turbulent lope flow In thi tudy, idealized turbulent anabatic and katabatic flow along double-infinite lope have been invetigated uing direct numerical imulation (DNS), which implie reolving all cale of turbulent motion down to the vicou diipation cale. The numerical algorithm employed to directly olve ()-(5) with Pr = ν / ν h = ha generally been the ame a the one ued to reproduce laminar [3-4] and turbulent [], [5] buoyantly driven flow along thermally perturbed vertical plate and loping urface. The value of the -3-3 urface buoyancy flux B have been taken a.3 m,.5 m for the katabatic flow cae and m,.5 m for the anabatic flow cae. Each flow type ha been invetigated with the lope angle α of 3º and 6º. For comparion, data from the Fedorovich and Shapiro [] tudy of turbulent flow along a vertical heated wall ( α = 9 ) are alo conidered. Applying the Reynold decompoition to the flow variable in the buoyancy balance equation (4) and averaging thi equation over time and patially over x-y plane parallel to the lope, we obtain: b b' w' Nu α + ν = in, (6) z z where prime ignify deviation from the average denoted by overbar. Integrating (6) over z from to and noting that both molecular and turbulent fluxe of the buoyancy vanih at, we obtain an integral form of the buoyancy balance: = /( in ) VL udz B N α, (7)

3 9 ème Congrè Françai de Mécanique Mareille, 4-8 août 9 where the product VL i the volume flux. Note that we do not define either the integral velocity cale V I or the integral length cale lope-flow Reynold number a: LI ( u / VI) dz. Baed on the above conideration, one may introduce an integral Re I VL / B ( N in ) Fp B / in ν = ν α α, (8) a) b) c) d) - FIG. Temporal variation of along-lope velocity component (u, plot a and c, in m ) and buoyancy (b, plot b - and d, in m ) at different level above the lope in the center of imulation domain for the katabatic flow cae ( B =.5 m, ν = m, N =, reulting in Fi = B 5 ) along lope of different teepne: 3 (plot a and b) and 6 (plot c and d). where the flow forcing parameter Fp B B ν N i negative for a katabatic flow ( B < ) and poitive for an anabatic flow ( B > ). The magnitude of Fp B repreent the ratio between the energy production at the urface and the work againt buoyancy and vicou force. From thi definition of Re I, we a particular lope flow to be more turbulent with increaing Fp B. In the performed DNS, value of Re I have been within the range of 3 to. 4 Reult of DNS The patial (in the z direction) and temporal evolution of the imulated velocity (u component) and buoyancy b field in the central point of the x-y plane i illutrated in Fig.. 3

4 9 ème Congrè Françai de Mécanique Mareille, 4-8 août 9 Reult are hown for two katabatic flow with the ame value of FpB = Bν N = 5, but with different lope angle: 3 and 6. After paing through relatively hort tranition tage, both flow become turbulent and diplay random, large-amplitude fluctuation of velocity and buoyancy field in the core region. At larger ditance from the lope, they how a quai-periodic ocillatory behavior. Notably, only fluctuation with a frequency equal to the natural buoyancy frequency N inα in the environmental fluid dominate at large z. Fluctuation with other frequencie rapidly decay away from the lope beyond the turbulent core of the flow. u (m - ) u (m - ) b (m - ) b (m - ) FIG. Mean along-lope velocity (u, olid line) and buoyancy (b, dahed line) profile in the katabatic (left) and anabatic (right) flow with B =.5 m, ν = m, and N = ( FiB = 5 ) for three different lope angle: 3 (blue line), 6 (red line), and 9 (black line). The mean velocity and buoyancy profile preented in Fig. reveal a coniderable enitivity of the mean flow tructure to the lope angle for both katabatic and anabatic flow cae. The mean profile were obtained by averaging the imulated flow field patially over x-y plane and temporally over at leat 7 ocillation period beyond the tranition tage. The hape of the katabatic-flow velocity profile for both lope angle < 9 are very different from the hape of velocity profile in the anabatic flow. The difference in hape between the buoyancy profile for both flow cae are le pronounced which i partially due to the fact that in both flow cae the buoyancy harply drop (increae) in the very cloe vicinity of the wall. In the katabatic flow, table environmental tratification in combination with negative urface buoyancy forcing lead to the effective uppreion of vertical turbulent exchange in the flow region in the immediate vicinity of the lope. A a reult, the mean velocity profile become jet-like, with a magnitude increae with decreaing lope angle, a alo exhibited in the Prandtl [] model. Depite the obviou tructural difference between the imulated katabatic and anabatic flow, the overall vertical extenion of the flow in term of the mean velocity appear to be almot the ame for flow along lope of the ame angle. A direct evaluation of the mean velocity integral in Fig. confirm the validity of (7) in thee flow cae. Therefore, the combination B /( N in α) = Fp Bν /inα of the governing parameter may be regarded a an integral dynamic imilarity contraint for lope flow forced by the urface buoyancy flux. It hould be noted that thi criterion applie to both turbulent and laminar lope flow. A revealed by the buoyancy variance profile in Fig. 3, the buoyancy variance in both lope flow cae attain their maxima extremely cloe to the wall, within the region of maximum gradient in the mean buoyancy profile (Fig. ). The velocity variance of notable magnitude are ditributed over layer that are typically a few time thicker than the layer which contain mot of the buoyancy variance. In both flow cae, narrow econdary maxima of uu ' ' in the cloe vicinity of the lope are oberved. Thee econdary 4

5 9 ème Congrè Françai de Mécanique Mareille, 4-8 août 9 maxima, which are not found in the flow along a vertical wall ( α = 9 ), become more pronounced with decreaing lope angle and are more prominent in the katabatic flow. u'u' (m - ) u'u' (m - ) b'b' (m -4 ) b'b' (m -4 ) FIG. 3 Slope angle dependence of the along-lope velocity variance (olid line) and buoyancy variance (dahed line) in the katabatic (left) and anabatic (right) flow preented in Fig.. 5 Summary Buoyantly driven lope flow along doubly-infinite cooled/heated inclined urface immered in a tablytratified fluid were numerically imulated by mean of DNS. The flow were driven by a patially-uniform urface buoyancy flux B and characterized by integral Reynold number in the range from 3 to. After the tranition from a laminar to a turbulent phae, the imulated lope flow entered quai-tationary ocillatory regime with frequency of ocillation (aociated with internal gravity wave) given by the product of the environmental Brunt-Väiälä frequency and ine of the lope angle, N inα. Turbulent fluctuation gradually faded out with ditance from the wall, while periodic ocillation perited within the outer laminar flow region before fading out. The lope-parallel velocity component variance conitently diplayed econdary maxima very cloe to the wall, at ditance comparable to thoe of the mean velocity maxima/minima. An integral dynamic imilarity criterion, /( in ) averaged thermal energy equation and quantitatively verified againt the DNS data. Reference udz = B N α, wa derived from the [] Fedorovich, E., and A. Shapiro, Turbulent natural convection along a vertical plate immered in a tably tratified fluid. Revied manucript ubmitted to J. Fluid Mech., 9. [] Prandtl, L., Führer durch die Strömunglehre, Vieweg und Sohn, Braunchweig, 38 pp., 94. [3] Shapiro, A., and E. Fedorovich, Prandtl-number dependence of unteady natural convection along a vertical plate in a tably tratified fluid. Int. J. Heat and Ma Tranfer, 47, , 4. [4] Shapiro, A., and E. Fedorovich, Natural convection in a tably tratified fluid along vertical plate and cylinder with temporally-periodic urface temperature variation. J. Fluid Mech. 546, 95-3, 6. [5] Shapiro, A., and E. Fedorovich, Corioli effect in homogeneou and inhomogeneou katabatic flow. Q. J. R. Meteorol. Soc., 34, , 8. 5