Note critical layer at z c where U(z)= c r, wave speed. In above profile z c is in shear stress layer (z c l s )for larger c r -as analysed for c i >0 by Miles(1957) Sajjadi S G, Hunt JCR, Drullion F (2013) Asymptotic Multi-layer analysis of wind over unsteady monochromatic surface waves J.Eng Maths
Waves and Interfaces Seminar at UCL Julian Hunt 1.Wind-Wave Coupling Introduction Physical mechanisms involved for wind over unsteady surface and wave groups. Triple(or quadruple) deck theory for turbulent shear flows. Combining sheltering and unsteady critical layer mechanisms. Explain why groups are most efficient mechanism for air-sea energy exchange. Through analytical and computational modeling.
Schematic Separated Sheltering L = O(a). Non-separated Sheltering. Schematic of wind over wave mechanisms for steady low amplitude waves. Shows sheltering mechanism in the surface layer and its coupling with the outer flow.
Steady waves (ci=0). Benjamin, Townsend, BH, Mastenbroek… non-separated sheltering. But critical layer is important ; drag is affected for c r >u*
Unsteady Waves For unsteady waves where 0 < c i /U* ~ 1. Mean streamlines of flow over the waves are shown in a frame moving with the waves. Waves closed loops are centred at the elevated critical height. Profiles of u–velocity perturbation for inviscid flow (when ν e = 0) is shown below. Perturbations become singular as the growth rate c I -> 0.
Waves growing; exp (i(k (x-c r t).exp(-k c i t). Inviscid flow Note singular behaviour as c i ->0, near critical layer In phase Vel perturbat ion Out of phase Vel perturb ation
Inviscid Mechanism for Unsteady Waves Fig of c I vs kz shows as c i 0 out-of-phase perturbation to \Delta u becomes very large. Occurs in very thin layer of thickness of O(c_i). Vorticity ( ω y) amplifies on lee-side and reduced on upwind-side. Leads to mean stream lines being deflected, "lower pressure" on the lee side "higher drag". when c i > 0, wave grows; c i <0 wave decays. But there is a net force on the wave produced by the integrated effect of the critical layer as |ci|->0. ( Miles did not calculate profiles-only the integral effect prop to U’’(at z=zc))
Historical Conclusions Miles (1957)/Lighthill (1962) calculate energy input/ growth for c i -> 0; a = const:; ka << 1. Note Lighthill’s analysis with delta functions (p192) suggests importance of thin, singular critical layers –but they were not mentioned explicitly. They conclude: there is a net inviscid force on monochromatic non-growing waves (steady waves).
Computations (Steady Waves) We adopt full realizable RSM (TCL) [Sajjadi, Craft & Feng]. We compare full Reynolds-stress model with DNS, Sullivan et al. Also adopted semi-implicit FD solver with LRR turbulence model for comparison. Waves closed loops are centred at the critical height. Very similar behaviour as DNS.
Computational Domain Schematic computational domain (84 x 50 x 5) mesh points. Bulk Reynolds number Re = 8000. Fully implicit, collocated, general curvilinear coordinates finite volume. Coupled to water through orbital velocities.
Energy Transfer from Wind to Waves (I) Total energy transfer parameter, \beta. Critical layer and sheltering mechanisms for unsteady waves (c i ) ( << U*) vs wave age c r /U 1. +++++, Miles (1957) calculation (c i = 0; ν e = 0) Thick solid line, Janssen (1991) parameterization of Miles (1957) for c i = 0 ; inviscid ; zero eddy viscosity Thin solid line, present formulation: (\beta_T + \beta_c) for c i >0 ; finite eddy viscosity. \circ, Numerical simulation using Reynolds-stress closure model for c i >0 ; finite Reynolds stress.
Monochromatic waves.Growth rate against wave speed / wind speed –models, and data Note how β rises with moderate increase in crit layer thickening of Inner viscous layer. But when cr/Ui ~5, the inner viscous layer diminishes the Sheltering mechanism, and β drops.
Monochromatic waves.Wave growth based on Miles model, with zero and finite eddy viscosity. Note no growth in latter case
Wave Groups (first proposed (?)by M.E.McIntyre –DAMTP ) On going project. Above application to group of waves Individual waves grow within group on upwind side and decrease on downwind side (hence plus/minus inertial critical layer effects). But shear layer over wave group leads to zc higher on lee side, U’’(zc) smaller. Hypothesis is net positive effect Study effect of non-uniform critical layer over groups (analytically & computationally). Preliminary results shown below.
Wave group -3 -5 waves; concept of double effect of sheltering + Inertial critical layer ; numerical simulations
Conclusions Physical growth mechanism is NOT soley due to steady Critical layer CL. Critical layer is unsteady problem. Growing/decaying wave amplitude in the groups increases CL on the downside of group where wave shape changes. CL plays an important role on sheltering. Asymmetrical sheltering leads to reduction in wind speed.
2. Tsunami waves The problem of negative tsunami waves travelling over long distances, being transformed into positive waves and impacting on beaches, before or after the transition. Ref.Klettner, C.A., Balasubramanian, S., Hunt, J.C.R., Fernando, H.J.S., Voropayev, S.I. and Eames, I., 2012 Draw down and run-up of tsunami waves on sloping beaches. Proc. Inst. Civil Engineers-Engineering and Computational Mechanics 165,119-129.
Earthqua ke off jap March 2011 Note depressio n to north
Measured data around the Pacific coast of Japan. Red and blue color bars indicate inundation height and run-up height. (a) Japan view, (b) Sendai Plain. Japan Tsunami (Neg wave to North of Sendai) Note maximum near Sendai Opposite Centre of Earth Quake. Note blue Inundation inland..
Negative Tsunami wave arriving at japan coast 2011
quaasi steady shallow water analysis c = sqrt (gh (x))(1 +/- ½ a(x.t) /h(x)) h(x)= ho –zs. (local water depth) Δc = ½ ([-/+ α w (x.t) – α b ] sqrt(g/ho) α b = dz s /dx. Let dz s /dt =+/- α w (x.t) Sloping water surface ; a(x,t)=+/-| α w (x.t) |( x f – x) -d| α w (x.t) |/dt = ½ ( +/- | α w (x.t) | - α b ) sqrt(g/ho)
SRI LANKA 2004—very large negative tsunami waves on beaches Caused very sudden and large loadings on buildings-
Backward Tsunami slope on the slope Where x< xmx, alpaw <0 Leading slope where alphaw <0 Surface slope ~ bottom slope. On trailing V shaped slop Wave slope |alphaw| tends to increase -> rapid change in rear wave shape -> shortening of wave dL/dt Larger amplitude wave, and force momentum at beach
Negative tsunami on a beach,tested in ASU hyd. lab Note the sharp rise on the downstream end behind the deepening depression-same n-l process as for + waves. Depression leads to shore line retreat large wave up the beach.
Note warming and less ice in Arctic ocean affects Wind driven Waves, which break up ice and leads to Further warming (Wadhams), and Less ice, plus seismic activity, is likely to Lead to tsunamis on very flat arctic coastline Concetn to local communities and to extractive industries And to possible transport dangers -
(a)Mean wind speed profiles at Delat Island tower in tropical cyclone Nuri (a)Mean wind speed profiles at Zhizai tower in tropical cyclone Hagupit Fig. 6 Mean wind speed profiles at the different radial positions (defined in Fig. 5) for the tropical cyclones Nuri and Hagupit. Note the pronounced low level wind speed maxima for the positions near the eyewall regions (FEW, BEW), and less pronounced in the outer regions (FOV, BOV) Up-down wave long after critical time. Note equal amplitude up=down. -note moving to right Wind structure 0f tropical cyclones using new data from met towers on south china coast. Liu, Kareem, Hunt
Mean wind profiles (a)Mean wind speed profiles at Delat Island tower in tropical cyclone Nuri (a)Mean wind speed profiles at Zhizai tower in tropical cyclone Hagupit Fig. 6 Mean wind speed profiles at the different radial positions (defined in Fig. 5) for the tropical cyclones Nuri and Hagupit. Note the pronounced low level wind speed maxima for the positions near the eyewall regions (FEW, BEW), and less pronounced in the outer regions (FOV, BOV)