A presentation given at the Isaac Newton Institute on 30th September 2022 by Frank Smith on new joint unpublished work with UCL PhD students T D Dang and Ellen Jolley

Dang, TD

Jolley, Ellen

Smith, Frank

English

UCL Discovery

On wall-ice accretion or melting in shear flowINI presentation 30th September 2022Frank Smith - UCL1. Background2. Pre-impact behaviour3. Droplets, impacts and ice growth4. Melting wall-ice in shear flowCollaborators:Richard Purvis – UEA Peter Hicks – AberdeenAndrew Ellis – UCLManolo Quero – Spain David Hammond – CranfieldPhil Wilson – New Zealand John Elliott – HullKevin Liu – UCL Ryan Palmer – Bristol Samire Yazar – TurkeyManish Tiwari – UCLAnd current UCL PhD students: Ellen Jolley Nat Henman T D Dang Qingsong LiuThanks to EPSRC, UCL, Totalsim, Aerotex1.   Background  / aircraft icing•Icing occurs during flight through clouds at or below freezing•Supercooled water droplets impinge upon forward facing parts of the vehicle•This water then freezes, either immediately on impact or after spreading back along the wingEngine icing.Modelling icingThere are many important parameters including:• Droplet size (or range of sizes)• Air Temperature• Air Speed• Water content of the cloud• Local geometry20 micron dropletsT=-6.6oCT=-12.2oCT=-16.6oCExisting models work well for small droplets (<40 microns)Droplet distortionFLOWSOLID, FLEXIBLE OR WATER-COVERED SURFACESkimming bodyLuxford & Hammond Hicks & FTSBody or ice particleDroplet onto roughnessEllis & FTSFTS et alALSO ALLOW FOR WATER LAYERS AND ROUGH SURFACES2. Pre-impact behaviour  -- e.g. particles in boundary layers Showing some recent and ongoing work on particles (Ellen Jolley, Ryan Palmer) and droplets (Nat Henman, Manish Tiwari)Thin particle travelling through boundary layer with Ellen Jolley 2022 JFMJFMParticle /body hits wallover this time scaleBody flies awayBody bounces repeatedlyAllowing for thin water layerAir flow  Water flow  Reduced equations in airReduced equations in water (different density)The body may fly away (as seen previously)Or it may skim (as below)Or it may hit the substrate and bounceOr it may sinkWorks with Ellen Jolley and Ryan Palmer With Nat Henman and Manish TiwariFree-surface solution for a droplet impact onto a surface (a) droplet free-surface (white), lubricant free-surface (yellow), velocity magnitude field (colour). (b) droplet free-surface (blue), lubricant free-surface (red), surface textures (black), with splash jet tip coordinates.• WATER:-div u = 0ut + (u.grad)u =  - gradp/ρ + g + ν(T) Δ uTt + (u.grad)T =  k Δ T• FREEZING:-Move ice boundary: change of state involving latent heatAllow a ‘nucleation layer’ of ice roughness at the bottom of the water layer• NUMERICAL METHOD:-Simulations use VOF / PLIC (piecewise linear interpolation calculation)3.  Droplets, impacts and ice growth Simulation for temperatureQuero et al (2006)Water droplet at – 10 degrees CWater film at 1 degree C50 ice crystals at – 0.1 degrees CThe crystals grow into each other and the ice front smoothsThe ice growth : -More analytical approach :Ice growth on a cold surface (post-impact)Small-time analysis as per Wagner et al, with viscous and thermal effectsWater dropletIce /roughnessSolid surfaceWaterfrom droplet                          Simple form : -Ice grows here Air or voidt < 0t > 0John Elliott & FTS There are many parameters : -Re (~ 104-5), Fr (~ 107), We (~ 105-7) σ (Prandtl, 7-13)St (Stefan, 10-2-10)β (kinetic underheating)Main findings are : -- icing can accelerate or decelerate the spreading of the droplet- multi-structure contains restricted regions of turbulence- parabolic shape of ice in certain conditions- connection with experimentsVariation of ice thickness (roughness) with time :   Symbols (experiments); red curves (maths predictions)ABCRough-ness(mm)Time (min)10630.54. Melting or growing wall-ice in shear flowWith T D Dang & Ellen Jolley Water flow u* = O(1) with shear           COLD WARM WALL ICE :  STEP-UP or HUMP FIG1 x* = x0* This study is on the melting or possible accretion of wall-mounted ice deep inside a boundary layer. FIG 1aThe configuration has water flow past a slender ice hump of relatively small finite length on a flat wall (or a step up), with the water being warmer than the ice.The wall is at the same temperature as the oncoming water except underneath the ice where the temperature of the wall is the same as that of the ice; a temperature-adjustment region may also be present in the wall just upstream and downstream of the ice hump. The aim is to find out if the whole ice hump melts, in what manner and how long it takes, and also whether accretion may take place instead.Non-dimensional Navier-Stokes and thermal equations:div u* = 0, Du* /Dt* = – grad p* + Re – 1 del2 u*,    Dθ* /Dt* = (σ Re) – 1 del2 θ* .  Here Re is the Reynolds number and σ is the Prandtl number. • The boundary conditions on the unknown ice surface y* = f*(x*, t*) lying between x* = x0* and x* = x1* say areu* + f*x*  v* = 0, v* – f*x*  u* = (1 – r*) f*t* ,  θ* = – 1Stefan condition on heat transfer ,  for x0* < x* < x1*.• On the fixed wall y* = 0 for x* < x0* and for x* > x1* we have the constraints of no slip and prescribed temperature,u* = v* = 0, θ* = c*(x*) .  The factor r* denotes the ice density relative to the water density. The prescribed wall temperature c* is assumed to tend to zero far upstream. It is also assumed to be continuous, implying in particular that c* = – 1 at the ends x0*, x1* of the ice hump. Far-field conditions on velocity, pressure and temperature associated with matching to the remainder of the flow field are also required in general.Parameter values:-Re (~ 104-5), Fr (~ 107), We (~ 105-7) σ (Prandtl number, 7- 8 for water at 18 degrees C)St (Stefan number, 10-2-10)For large Reynolds numbers the scenario of interest occurs close to the wall within a boundary layer whose length and height scales are of O(1) and Re – ½ respectively whereas the length scale of the ice hump (or its front) is small. Locally the ice surrounds the position x* - x0* = Δ x  close to the wall:-A local scaling of velocity, pressure, temperature, space and time applies.  Local flow u = O(1) with uniform shear           COLD WARM WALL ICE :  STEP-UP or HUMP FIG1b x = 0 Now the governing equations reduce toux + vy = 0 ,  uux + vuy = – px(x, t) + uyy ,  uθx + vθy = σ – 1 θyy ,  with p = p(x, t) being independent of y because of the normal momentum balance. The boundary conditions becomeu – λy → 0, θ  – φy → 0 as y → ꝏ ,  u = v = 0, θ = c(x) at y = 0 for x < 0 or x > 1 ,  u = v = 0, θ = – 1 at y = f(x, t) for 0 < x < 1 ,  B ft = – θy at y = f(x, t) for 0 < x < 1 .  • B = 1 without loss of generality.    • The given function c(x) is the scaled form of the wall temperature and is continuous, with c(0) = c(1) = – 1 and c(ꝏ) = c( – ꝏ) = 0. The constant factors λ, φ are present because of the locally uniform shear flow and heat transfer from the boundary layer above the local region. • The far field here has (u, v, p, θ) = (λy, 0, 0, φy) and, although λ is positive for the current fluid flow from left to right, the heat transfer contribution φ can be positive or negative. • The local flow and thermal balances in (2.5b, c) are quasi-steady due to the assumption of slow erosion of the ice hump. Ours is a basic case: no kinetic underheating, etc. The aim is to determine the flow and temperature properties and, especially, the evolution of the ice shape and the wall shear. The lack of upstream influence provided the flow is forward (u positive) somewhat simplifies the task of solving. On the other hand algebraic decay is found in the temperature field at large y which somewhat complicates the task. We seek an analytical solution first for small disturbances to the far-field flow. Linearized solution for small heights.If the ice height f is small, say f = hf1, (u, v, p) = (λy, 0, 0) + h(u1, v1, p1) + …, θ = O(1) .  THIS LEADS US TO A LINEAR SYSTEM:-u1x + v1y = 0 ,  λy u1x + v1 λ = – p1x (x, t) + u1yy ,  λy θx = σ – 1 θyy ,  subject to u1 → 0, θ  – φy → 0 as y → ꝏ ,  u1 = v1= 0, θ = c(x) at y = 0 for x < 0 or x > 1 ,  u1 = – λf1, v1 = 0, θ = – 1 at y = 0 for 0 < x < 1 ,  f1t’ = – θy at y = 0 for 0 < x < 1 ,  with t set to hBt’ with t’ of O(1), corresponding to faster evolution now.The solution for the thermal behaviour is obtainable through a Fourier transform in x, giving(θ  – φy) (FT) = C(FT) Ai( (i α λ σ) 1/3 y) ,  f1t’ = RHS , where (FT) denotes the transform, α is the transform variable and Ai is the Airy function. The function C(x) is c(x) for x < 0, x > 1 and – 1 for 0 < x < 1, and RHS = – φ – Ai′(0) (λ σ) 1/3   ʃ(a, x) (x – s) – 1/3 c ′ (s) ds  / (Ai(0) Γ(2/3)) ,  with x restricted to 0 < x < 1. (For other x values, the function – STUFF determines the scaled heat transfer at the wall.) Hence f(x, t’) = f(x, 0) + (RHS) t’ if f>0, and f(x, t’) = 0 otherwise,  gives the ice shape, with allowance made for complete melting.In this linearized regime the background fluid flow determines the coefficients λ, φ and these then determine the thermal response, which then determines the shape evolution, which then determines the flow perturbation.Nonlinear solution for medium heights.Use is made of the Prandtl transposition in which y = f + y**, v = fxu + v**, leaving (2.5a-c) intact except for the replacement of y by y** and v by v**. The boundary conditions become  (ignoring asterisks)u – λy → λf, θ  – φy → φf as y→ ꝏ , u =  v =  0, θ = c(x) at y = 0 for x < 0 or x > 1 , u =  v =  0, θ = – 1 at y = 0 for 0 < x < 1 , B ft = – θy at y = 0 for 0 < x < 1 , Solutions are shown below, with σ = 5 and φ = 0.StepsErosion point agrees with linear resultHumpsVelocity and temperatureprofiles for case h = 1Shows flow reversaland eddy formationFurther analysesFor steps at large times, xerosion ~ t3Large σmulti-structureFor any icing shapes, small t origin shift of erosion pointFinal behaviour for step and hump casesSmall-time behaviour.Consider the early behaviour for the basic case of no background heat transfer: (u, v, θ, p, f) = (u0, v0, θ0, p0(x), f0(x)) + t (u1, v1, θ1, p1(x), f1(x)) + … .  Here the subscript 0 denotes the initial state of flow and temperature over the initial ice hump F0(x). The perturbations with subscript unity then satisfy a linear system. In particularf1 = – θ0y at y = 0,confirming that the main thermal properties control the temporal change in the ice shape initially. Of interest is the response near the beginning of the ice. There f0(x) is (say) linear in x whereas typically f1(x) is O(x -1/3 ) , since θ is O(1) from the surface condition and y is of order x 1/3. Hence the ice shape takes the form O(x) + O(x -1/3 t) locally, implying that there exists a small subzone close to the beginning where x is O(t 3/4). In the subzone, x = t 3/4 x~, y = t ¼ y~, u = λ t ¼ y~ + t ¾ u~ + …, p = … , f = t 3/4 f~(x~), θ = O(1) , The thermal equation becomes or remains λy θx = σ – 1 θyy ,subject to matching and boundary conditions. A similarity solution holds,θ = q(η) ; η = y~ / x~ 1/3  f~ = a0 x~  – x~  - 1/3 q’(0) Erosion point is :                           x = (q’(0)/a0) ¾ t ¾ for small times.Final behaviour (vanishing) of ice lumpsHump is then small   linear effects   y ~ x 1/3 over entire hump but history matters  θy (y = 0) = g(x) ~ 1 and g(x) stays the same for all late t. ft = - g(x)  for some late time t > t1.Hence if f(x, t) has shape f1(x) at time t=t1 say thenf(x, t) = f1(x) – (t – t1) g(x) :- For parabola, vanishing point x = 4/7.g(x)f1(x)Final behaviour (spreading erosion) of ice stepsFindf ~ 1 – A t x – 1/3 where A = 3 (λ / 9σ)1/3 /  Γ(1/3)Spreading downstream is given by x ~ t 3Strongly nonlinear solution for large heights.This tells us what happens for much thicker ice. Suppose φ is of order unity. If f = hF with h >> 1, a two-zoned flow structure emerges. OUTER ZONE: y** = hY+ andu = h (λY+ + λF), ψ = h2 (½ λ Y+ 2 + λFY+ ), p = - ½ h2 λ2 F2 .---- an exact solution of the governing equations which satisfies the outermost boundary conditions but leaves a nonzero slip velocity u ~ hλF at Y+ = 0+. The corresponding thermal solution is θ = h φ (Y+ 2 + 2 F Y+ ) ½ .The algebraic decay into the far-field response φhY+ at large Y+ is notable. This outer zone has inviscid motion.INNER ZONE : thin layer on the ice, with y** = h – ½ Y and (u, v**, θ) = (hU, h ½ V, h ¼ Θ) + … .The time scaling is t’ = h ¼ T, implying that the evolution is relatively slow. Substitution into the governing equations yieldsUx + VY = 0 ,  UUx + VUY  = λ2 F Fx + UYY ,  UΘx + VΘY = σ – 1 ΘYY .  The boundary conditions :U → λF(x, T), Θ ~ φ (2 F Y) ½, as Y → ꝏ ,  U = V = 0, Θ = 0, FT =  – ΘY, at Y = 0.  • This is a classical boundary-layer problem with the pressure effect prescribed at each time level T, supplemented by the temporal evolution of the scaled ice shape F. • The ice temperature – 1 is negligible to the present order. • The square-root trend with Y is unusual.The above formulation holds on the iced part of the surface, not upstream since the scaled temperature θ is in effect zero there. Expect positive background heat transfer φ > 0 to produce positive heat transfer on the ice at Y = 0, thus inducing ice erosion, whereas negative φ might provoke negative heat transfer on the ice, inducing ice growth, but in any case the nonlinearity of the system and the x-dependence need to be taken fully into account. Numerical solutions are being considered.If φ is small or zero:-The critical scale here is φ = h-1/4φ^ because it reinstates the influence of the ice temperature at leading order in the inner zone. It givesU → λF(x, T), θ ~ φ^ (2 F Y) ½, as Y → ꝏ , U = V = 0, θ = – 1, FT =  – θY, at Y = 0.The presence of ‘– 1’ is noted.ConclusionsPre-impact models have been developed for ice particles passing through an air boundary layer or through air and water.Post-impact ice accretion due to a droplet spreading on a wall has been described.Wall-ice melting or accretion in near-wall flow is being studied, beginning with a simple model.Further work is ongoing. Hot-wall effects; rebounds; viscosity; 3D.

On wall-ice accretion or melting in shear flow

https://discovery.ucl.ac.uk/10156742/1/INI%20Sept%202022%20%28Frank%20Smith%29.pdf

On wall-ice accretion or melting in shear flow

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