HYDRODYNAMIC STABILITY ANALYSIS OF SHEARED CONVECTIVE BOUNDARY LAYER FLOWS IN STRATIFIED ENVIRONMENTS

une 3 - uly 3, 15 Melourne, Australia 9 P-51 HYDRODYNAMIC STABILITY ANALYSIS OF SHEARED CONVECTIVE BOUNDARY LAYER FLOWS IN STRATIFIED ENVIRONMENTS Y. Xiao, W. Lin, Y. He College of Sccience, Technology and Engineering ames Cook University Tonsville, QLD 811, Austriala email: yuan.xiao@my.jcu.edu.au email: enxian.lin@jcu.edu.au email: yinghe.he@jcu.edu.au S. W. Armfield, M. P. Kirkpatrick School of Aerospace, Mechanical and Mechatronic Engineering The University of Sydney NSW 6, Australia email: steven.armfield@sydney.edu.au email: michael.kirkpatrick@sydney.edu.au ABSTRACT Hydrodynamic staility analysis is carried out on sheared convective oundary layer (SCBL) flo, in hich oth sheared stratified flo and thermally convective flo coexist. The linear unstale stratification for the thermal convective flo region is included in the Taylor-Goldstein equations in terms of an unstale factor in parallel ith a stale stratification factor. Ne unstale regions corresponding to the thermally convective instaility are found aove the pure sheared stratified flo regions in the avenumer α versus stratification factor plane. As the stratification ratio / increases, the unstale regions of thermal convection gradually approach and dominate the shear stratified unstale regions. The transition from shear stratified unstale mode to thermal convection unstale mode is also oserved in the temporal groth rate σ versus plane and in the σ versus α plane. The eigenfunctions of uoyancy, vertical velocity and uoyancy flux perturations for sheared stratified dominant mode, the transitional mode and the thermal convection dominant mode in SCBL flo configuration are also discussed. INTRODUCTION As a representative geographical flo, sheared stratified (SS) flo in hich sheared flo motions occur in a stratified environment has received sustantial studies in the past decades. Hoever, in the geographical settings such as the planetary or oceanic oundary layers and engineering settings involving heat transfer from radiation or chemical reaction, the comparale thermally convective flos inevitaly coexist ith sheared stratified flo, together forming a complex flo configuration, namely the sheared convective oundary layer (SCBL). SCBL flos have significant importantce for environmental issues, such as the mixing process of pollutants in atmospheric oundary layers, the heat and mass transfer in the upper ocean, and the mixing in large scale ater odies such as reservoirs, lakes and estuaries. The SCBL flos in fire-induced smoke transportation ill potentially increase the fire haards y changing flo patterns. As pure thermally convective flos and pure sheared stratified flos have independent unstale modes, i.e., the Rayleigh-Benard instaility and shear stratified instaility (e.g., the Kelvin-Helmholt instaility and Holmoe instaility), the comination of the to asic flo configurations are expected to produce more interactive and complicated unstale modes and corresponding flo patterns. For instance, Raasch & Franke (11) used high resolution numerical simulation to find the transitional ehaviour from a spoke-shape flo pattern in pure thermal convection to a and-like flo pattern in sheared thermal convection. Yang et al. (1) oserved experimentally a unique flo pattern near the interface in fire induced SCBL. Most recently, Steart et al. (1) investigated the SCBL flos ith todimensional direct numerical simulation and found that the Kelvin-Helmholt instaility and the Rayleigh-Benard instaility coexist in some stratification conditions, here the presence of the penetrative convection modifies the original Kelvin-Helmholt eddy structures. Their results also suggest that ne unstale modes might occur in the SCBL flo settings, hich motivates the current hydrodynamic staility analysis to examine the staility features of the SCBL flo. This paper ill study the hydrodynamics of SCBL flo y introducing an unstale stratification factor into the Taylor-Goldstein equation that descries the hydrodynamics of shear stratified flo. This idea comes from the hydrodynamic study of to-layer thermal penetrative convective flos here a thermally convective region is capped 1

y a stale stratified layer. The difference eteen the current SCBL flo and such to-layer penetrative convection is hether horiontal shear flo is present. By solving modified Taylor-Goldstein equations for SCBL flos, the instaility features of the SCBL, including temporal groth rate and spatial perturation structures, ill e investigated, after a rief introduction of the numerical methodology. PERTURBATION EQUATIONS Based on the Taylor-Goldstein equations for sheared stratified flo and y applying the Squire transformations as detailed in Drain & Reid (), the folloing perturation equations, ritten in matrix form, have een derived y the current study, σ [ s ][ ] = i α(u I ŵ α [ ] s U ) Fr ŵ Ñ i αu (1) Figure 1. The SCBL flo configuration under consideration and the ackground velocity and stratification profiles. here the suscript denotes the second order differentiation ith respect to, s is the Squire Laplacian operator defined as s = D α, D = / is the differential operator for the perturation properties, α is the Squire avenumer, I is the identity matrix, σ is the Squire temporal groth rate of the perturations properties, ŵ and ( = γ θ) are the vertical component of the velocity perturation and the Squire uoyancy, θ is the Squire temperature, γ is the thermal expansion coefficient, and Ñ (Ñ = γ θ ) is the local Squire uoyancy Brunt-Väisälä frequency. For sheared stratified flos, the sheared layer thickness, the velocity and temperature changes across the sheared/stratified layer are usually selected as the characteristic length scale L, velocity scale u, and temperature scale θ,, respectively. Fr = u, / gl is the Froude numer here g is the gravitational acceleration. Hael (197) suggested that hen the asic velocity and ackground stratification in the sheared/stratified layer are in the form of u, f ( ) and θ, f ( ), here f ( ) is a hyperolic function, and if ( u/ ) = = 1 and ( θ/ ) = = 1, here = is the central line of the sheared/stratified layer, then the local Richardson numer Ri g () is layer is G s = θ, /L, here the positive sign indicates that the temperature increases ith increasing height. In penetrative convection prolems here a thermal convection region is capped y a staly stratified layer, Whitehead & Chen (197) and Sun (1976) introduced a staility factor S y replacing the ulk temperature gradient in the Rayleigh numer ith the ulk temperature gradient in the top staly stratified layer. Inspired y this method, in the current SCBL flo, e replace the ulk temperature gradient G s in ith G in the convective flo region < L and define the unstale stratification factor, for SCBL flo as = gγ θ, L ( ) L = gγg L u ( ) = G, (3), u, G s so that hen > L, N = ( θ/ ) and hen < L N = ( θ/ ) in (). It is noted that and have the opposite signs, indicating unstale and stale ackground stratification, respectively. Therefore, hile solving (1), N ill ecome a pieceise function depending on if > L or < L. Ri g () = N ( ) ( u / ) = ( θ/ ) ( u/ ), () in hich = gγ θ, L/ u, = (gγ θ,/l)(l/ u, ). Here the ulk temperature gradient θ, /L ithin the sheared/stratified layer is extracted. In practical staility analysis, plays as an effective sustitiute for Ri g. The SCBL flo system in this study is shon in figure 1, here the convective flo region ith unstaly linear stratification and the length scale L occurs at the ottom of the sheared stratified flo region ith smooth tangential stratification and the length scale L. As the temperature at the ottom oundary θ, is larger than the temperature θ 1, the flos are unstale at < L. Hoever, hen > L, the flos are staly stratified as θ, = θ, θ 1, >. In the convective flo region the ulk temperature gradient is G = θ, /L, here the negative sign indicates that the temperature decreases ith increasing height. Correspondingly, the ulk temperature gradient for the staly stratified METHODOLOGY The temporal mode of (1), here α is a real numer and c is a complex numer, is solved in this study. It should e noted that the real part of the temporal groth rate, σ = i αc, of the perturation is only determined y the imaginary part of the complex ave speed c, for the avenumer α is fixed as a real numer. Matrix methods are used to solve the eigenvalue prolems formed y discretising the perturation equations (1) ith uniform grid and using the second-order central difference scheme. The QZ algorithm developed y Moler & Steart (1973), hich is integrated in the LAPACK routine CGGEV, is used as the complex eigenvalue solver. The roustness of the QZ algorithm in hydrodynamic staility analysis has een demonstrated in some recent studies, such as Smyth et al. (11), Liu et al. (1), and Thorpe et al. (13). The oundary conditions ŵ = = are applied at oth the top and the ottom oundaries. The dimensionless verti-

(a). = =.1 =. 1.5 =.3 1. =. =.6.5 =.8...5 1. 1.5. () = (1/6). 1.5 1..5...5 1. 1.5. (c) = (1/3). 1.5 1..5...5 1. 1.5. Figure. Contour plots of the real part of temporal groth rate σ in the versus Squire avenumer α plane ith (a) =, () = (1/6) and (c) = (1/3). The magnitudes of σ are denoted y curves in different colors. cal coordinate varies eteen -5 and 5, here is made dimensionless y δ s, giving the computational domain a sie ten times that of δ s, here the characteristic length δ s is selected as one half of the sheared layer thickness. The length scale for the unstale stratified layer is L = 3δ s, hich allos the unstale stratified layer to e adjacent to the central shear stratified layer. Based on (3), hen θ, = θ,, = G = (1/3). Thus, this study ill change in the ay that / ill e the factor of 1/3. RESULTS Groth Rate σ Figure shos the contours of the real part of the temporal groth rate σ in the α plane for different unstale stratifications in terms of. In figure (a) here = hich represents that no convective flux exists, the unstale hemi-ellipse regions over σ = 1, in hich the imaginary part of σ (Im[ σ]) is found to e ero, correspond to the stationary Kelvin-Helmholt instaility mode in SS flo as marked y SS. The shapes of the KH mode regions are similar to the numerical results from Hael (197), here the critical stratification factor cr occurs at a corresponding critical avenumer α cr =.5. As hen a convective flux is introduced, the other large unstale region here Im[ σ] appears, as marked y RB in figure () and figure (c). In figure (), the RB region interacts ith the SS region at α.8. In figure (c) hen is one thrid of, the intersection points move to the small avenumer range α.5 and the to contour curves of α =. merge together at α =.5.8. Inside this merging region, part of the σ =.1 contour, hich as in the SS..15.1.5 SS Transition RB =(/3) =(1/3) =(5/3) =(1/3). =(1/6)....6.8 1. Figure 3. Calculated real part of the temporal groth rate σ plotted against at α =.5. The results in the SS and RB unstale regiosn are denoted y the dashed and solid curves, respectively. region, is overhelmed y the σ =. contour of the RB region. To further study the interactions eteen the to instaility regions, figure 3 shos Re[ σ] plotted against at the critical avenumer α cr =.5 for a series of / ratios. This figure can e considered as a vertical slice plot of figure at α =.5. The Im[ σ]= solutions for the SS mode and the Im[ σ] solutions for the RB mode are denoted y dashed and solid curves, respectively. It is found that as different / are applied, the SS curve retains its original shape at = although different RB curves are otained. Therefore, the SS mode is independent of the RB mode. As the SS mode ranch decreases quickly ith increasing and the RB mode ranch increases gradually ith increasing, the to ranches intersect near the critical stratification factor =.5 for SS flo. Near the intersection point, as Re[ σ] in the RB mode is comparale to that in the SS mode, a narro transition region here oth SS and RB have equal influence is created. As / increases, the RB mode curves gradually rise upards and cap the SS mode curves, therey moving the intersection point upards as ell. As a result, some unstale regions hich ere in the SS mode are replaced y the RB mode, e.g., =..5 elong to the SS region for < (1/3) ut elong to the RB for > (1/3). Figure shos the dispersion relations eteen Re[ σ] and avenumer α at =.. It can e considered as a horiontal slice plot of figure at =.. Similar to the σ plot in figure 3, the dispersion curves for the SS and the RB modes are represented y dashed and solid curves, respectively. Different from figure 3, to intersection points eteen the SS mode dispersion curve and each RB mode dispersion curve are found at small avenumer range and large avenumer range respectively, forming to transitional regions in the α σ plane. As increases, the RB dispersion curves gradually rise ased on their common origin point and move the to intersection points to larger Re[ σ]. Consequently, more and more the SS mode dispersion curve is dominated y the RB mode dispersion curve, e.g. hen = (8/3) most parts of the SS dispersion curve are covered y the RB curves. To quantify ho the RB mode gradually overhelms 3

Re[ ].8.6.. =(8/3) =(5/3) =(/3) =(1/3).....6.8 1. Figure. Calculated real part of the temporal groth rate σ plotted against α at =.. The results in the SS and RB unstale regiosn are denoted y the dashed and solid curves, respectively. (a) =.1 - - -.5..5 (c) =. SS - - -.5..5 - - -.5..5 (d) = 1. - - () =. RB -.5..5 Figure 6. Calculated eigenfunctions of the uoyancy at α =.5 ith (a) =.1, () =. for the RB ranch, (c) =. for the SS ranch, and (d) =1.. The solid and dashed curves represent the amplitude and the phase, respectively. Re[ ].5..3. / Figure 5. Calculated real part of the temporal groth rate σ plotted against the stratification ratio / at =. and α =.5. The results in the SS and RB unstale regiosn are denoted y the dashed and solid curves, respectively. The paraolic correlation for the RB mode plot is denoted y the red solid curve. the SS mode as increases, figure 5 shos Re[ σ] plotted against the stratification ratio / for oth the SS and the RB modes at α =.5 and =.. As oth and α are fixed, Re[ σ] for the SS mode is constant at.5. On the other hand, Re[ σ] for the RB mode increases ith increasing / in a paraolic fashion, ith the folloing correlation, Re[ σ] RB =.1( ) +.165( )+.837, () here the suscript RB indicates that the groth rate elongs to the RB mode. It is noted that the RB mode curve and the SS mode curve intersect at / = 3, hich indicates a critical transition stratification condition from the SS mode to the RB mode. Eigenfunctions Based on figures 3 and 5, the eigenfunctions for uoyancy perturation, vertical velocity perturation in (1) and their product, and uoyancy flux perturation B = θ are studied at the critical avenumer α =.5 and stratification ratio / = 3, hich is the intersection point found in figure 5. Three typical values are selected at =.1 here the SS mode dominates, at =. here the transition from the SS mode to the RB mode occurs, and at = 1. here the RB mode dominates. For =., the Re[ σ] for the SS and RB modes are very close to each other, therefore the eigenfunctions for oth modes are shon. Figure 6 shos the calculated eigenfunctions for at α cr =.5 and / = 3 ut ith three typical values as discussed aove. The solid and the dashed curves represent the real and the imaginary parts of, hich correspond to the amplitude and the phase of, respectively. At =.1 here the SS mode dominates, concentrated on the initial central shear/stratified layer, despite slight deviations of phase (dashed curve) due to eakly convective flo induced y the ottom unstale stratified layer. As increases to., for the RB mode as shon in figure 6(), varies drastically near =, hich is the vertical height of the interface eteen the initial stale and unstale stratified layer. For the SS mode as shon figure 6(c), the variation of remains at the central shear layer ut its amplitude ecomes positive compared to the negative one as shon in figure 6(a). As increases to 1. here the RB mode dominates, the asolute magnitude of the phase change (dashed curve) evolves to e larger than the amplitude change (solid curve) of at =. Meanhile, the amplitude variation along in figure 6(d) keeps a similar profile to that shon in figure 6(), and also shos a similar profile to that Sun (1976) here only penetrative convection occurs. Figure 7 shos the calculated eigenfunctions for at α cr =.5 and / = 3 ut ith three typical corresponding to figure 6. At =.1, the central line of amplitude of is located slightly aove the center of the initial shear layer ( = ), as a result of eak convection flo due to a small. At =., the strong variation section of starts from the ottom of the domain for oth the SS and RB modes.

- - -1. -.5..5 1. - - (a) =.1 (c) =. SS -1. -.5..5 1. () =. RB - - -1. -.5..5 1. (d) = 1. - - -1. -.5..5 1. Figure 7. Calculated eigenfunctions of the vertical velocity perturation at α =.5 ith (a) =.1, () =. for the RB ranch, (c) =. for the SS ranch, and (d) =1.. The solid and dashed curves represent the amplitude and the phase, respectively. - - -. -.... (c) =. SS - - (a) =.1 -. -.... - - -. -.... (d) = 1. - - () =. RB -. -.... Figure 8. Calculated eigenfunctions of the uoyancy flux at α =.5 ith (a) =.1, () =. for the RB ranch, (c) =. for the SS ranch, and (d) =1.. The solid and dashed curves represent the amplitude and the phase, respectively. For the RB mode, as shon in figure 6(), the profile of the amplitude of changes remarkaly at = 5 1, indicating that most of originate from the ottom convective flo rather than from the initial shear layer hich extends from = 1 1. For the SS mode as shon in figure 6(c), although the amplitude of starts to vary from the ottom of the domain, the maximum still occurs ithin the initial shear layer here = 1 1. At = 1. here the RB completely dominates, shos strong propagative features as the magnitude of phase is larger than the magnitude of amplitude. Figure 8 shos the calculated eigenfunctions for the uoyancy flux perturation B = at α cr =.5 and / = 3 ut ith three typical corresponding to figures 6 and 7. At =.1, the uoyancy flux perturation is formed on the initial shear layer at the center of the domain. As increases to., significant differences are found eteen the SS and RB modes. For the RB mode, as shon in figure 8(), strong uoyancy flux perturation occurs near the interface at =, ith intense and positive magnitude of amplitude compared to figure 8(a). For the SS mode as shon in figure 8(c), although the variations of B occur mainly in the initial shear layer, such variations appear very eak as the magnitude of the amplitude and phase significantly decrease compared to figure 8(a) and 8(). At = 1., the uoyancy flux perturations are almost negligile. The transitional ehaviors of B from the SS mode to the RB mode suggests that as unstale stratification increases, the uoyancy flux generated from the ottom thermal oundary ill gradually smooth out the uoyancy flux perturations generated y the central shear stratified layer and eventially dominates in the entire domain. CONCLUSIONS The influences of the ottom thermal convection region, in terms of an unstale stratification factor, are added to the Taylor-Goldstein equations to descrie the staility features of the sheared convective oundary layer (SCBL) flo. As is introduced into the Taylor-Goldstein equation system, ne unstale regions indicating thermal instaility are found aove the critical stratification factor =.5, for the shear stratified flo, in the α plane, in addition to the semicircle instaility region of the classic sheared stratified flo at <.5. As the stratification ratio / further increases, the thermal unstale region expands and gradually merges ith the shear stratified unstale regions. In the σ and α σ planes, increasing expands the thermal unstale regions and accordingly leads to domination of the thermal unstale mode over the shear stratified unstale mode. It is further found that the temporal groth rate of the thermal unstale mode increases in a paraolic fashion ith the stratification ratio / and gradually approaches, intersects and at last overhelms the temporal groth rate of the shear stratified mode, leading to a transition from shear stratified dominant SCBL mode to thermal convection dominant SCBL mode. The critical transition condition for SCBL flo is found at / = 3 and α =.5. The analysis of the eigenfunctions of the uoyancy perturation, vertical velocity perturation and uoyancy flux perturation B = shos distinctively different spatial perturation structures for the shear stratified dominant mode, transitional mode and the thermal dominant mode. ACKNOWLEDGEMENT The support from the Australian Research Council is gratefully acknoledged. Y. Xiao also ould like to thank ames Cook University for the CUPRS scholarship. REFERENCES Drain, P. G. & Reid, W. H. Hydrodynamic Staility, nd edn. Camridge University Press. Hael, P. 197 Numerical studies of the staility of inviscid stratified shear flos. ournal of Fluid Mechanics 51, 39 61. Liu, Z., Thorpe, S. A. & Smyth, W. D. 1 Instaility and hydraulics of turulent stratified shear flos. ournal of Fluid Mechanics 695, 35 56. Moler, C. B. & Steart, G. W. 1973 An algorithm for generalied matrix eigenvalue prolems. SIAM ournal on Numerical Analysis 1, 1 56. 5

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