When a semi-infinite body is heated from below by a sudden increase in temperature (or cooled from above) an error function temp erature profile grows as the heat diffuses into the fluid. The stability of such a profile is investigated using a large-wavelen gth asymptotic expansion under the quasi-static, or frozen-time, approximation. The critical Rayleigh number for this layer is f ound to be Ra=√(1/2) based on the length-scale √(κ t) where κ is the thermal diffusivity and t the time since the onset of heating

Kerr, O.

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Kerr, O. (2016). Critical Rayleigh number of an error function temperature profile with a quasi-static assumption. City Research OnlineOriginal citation: Kerr, O. (2016). Critical Rayleigh number of an error function temperature profile with a quasi-static assumption. Permanent City Research Online URL: http://openaccess.city.ac.uk/17472/ Copyright & reuseCity University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact with the author(s) of this paper, please email the team at publications@city.ac.uk.Critical Rayleigh number offor error function temperature profilewith a quasi-static assumptionOliver S. Kerr∗Department of Mathematics, City, University of London,Northampton Square, London, EC1V 0HB, U.K.(Dated: September 16, 2016)AbstractWhen a semi-infinite body is heated from below by a sudden increase in temperature (or cooledfrom above) an error function temperature profile grows as the heat diffuses into the fluid. Thestability of such a profile is investigated using a large-wavelength asymptotic expansion under thequasi-static, or frozen-time, approximation. The critical Rayleigh number for this layer is found tobe Ra = pi1/2 based on the length-scale (κt)1/2 where κ is the thermal diffusivity and t the timesince the onset of heating.PACS numbers: 47.20.Bp, 47.55.P-∗Electronic address: o.s.kerr@city.ac.uk1I. INTRODUCTIONUnderstanding when an evolving system starts to become unstable is not always straightfroward. For example, when a semi-infinite body of fluid is heated from below by impulsivelyincreasing the bottom temperature by ∆T at an initial time t = 0 then the heat diffusesinto the fluid resulting in a temperature profileT = T0 + ∆T erfc(z2(κt)1/2), (1)where T0 is the initial temperature, κ the thermal diffusivity and z the distance from thewall. Intuition may be sought by looking at the region of greatest temperature gradientsdriving the instability, and making a comparison with results from the case of a layer betweenboundaries with comparable width and gradients. The stability of a horizontal layer of widthD with an imposed temperature difference of ∆T is determined by the Rayleigh numberRa =gβ∆TD3νκ. (2)where g is the acceleration due to gravity, β the coefficient of thermal expansion and ν thekinematic viscosity. One may think of D as growing like (κt)1/2. This would lead one toexpect the system to start off as stable and become unstable as the instantaneous Rayleighnumber exceeded some critic value. An estimate of this value may be found by a quasi-static or frozen-time analysis where the evolving temperature profile is assumed to be fixed.This approach was used by Currie [1] who used a piece-wise linear approximation to thetemperature profile. Using a length-scale l chosen so that the added heat in the fluid is thesame as that of the error function profile, the critical Rayleigh number for a semi-infinitefluid was found to be 32 for a no-slip boundary. In comparison, for the case of a lineartemperature gradient in a layer between two horizontal boundaries the critical Rayleighnumbers are 27pi4/4 for stress free boundary conditions and 1707.8 for no-slip boundaries[see, for example, 2]. However, the quasi-static analysis of the error-function profile givenby (1) seems not have been done. The purpose of this paper is to rectify this omission.As the instability for the semi-infinite fluid is a long-wave instability we will look atthe large along-wall wavelength asymptotic limit. The solution separates into two parts.One part is near the wall on the length-scale of the thermal diffusion distance (κt)1/2. Theother part represents the adjustment of the bulk of the fluid to the perturbations in this2             T0T0 + ∆T(κt)1/2FIG. 1: Schematic diagram showing the temperature profile for the heating from a single lowerboundary.boundary layer and is on the length-scale of wavelength of the disturbances. We will matchthe solutions in these regions using the method of matched asymptotic expansion [3, 4] andhence show the critical value of the Rayleigh number is Ra =√pi.II. ASYMPTOTIC ANALYSISHere we look at the heating of a semi-infinite body of fluid from a single horizontal bound-ary below the fluid. In this situation there is a growing destabilizing temperature gradientof height of order (κt)1/2 where t is the time since the onset of heating and κ the thermaldiffusivity. This is shown schematically in figure 1. If the boundary temperature is increasedto T0+∆T and held at this level the background temperature profile, T (z, t), is given by (1).We will restrict ourselves to looking at incompressible, two-dimensional motions, and so wecan use the vorticity–streamfunction formulation. We can derive nondimensional equationsusing the scalings of D for length, D2/κ for time, ∆T for temperature, κ/D2 for the vor-ticity and κ for the streamfunction where D is a suitable length-scale. In this quasi-staticor frozen-time analysis we use the length-scale D = (κt)1/2, and the time variation of D isignored and it is taken to be a constant. This gives the linearized nondimensional equationsfor the vorticity, ω streamfunction, ψ and temperature, T ,1σ∂ω∂t= −Ra ∂T∂x+∇2ω, (3a)∇2ψ = −ω, (3b)∂T∂t+∂ψ∂x∂T∂z= ∇2T. (3c)3The Rayleigh number, Ra, is the nondimensional measure of the heating, defined previouslyby (2), and the Prandtl number is σ = ν/κ. The nondimensional background temperaturegradient is given by∂T∂z= −e−z2/4pi1/2. (4)The boundary conditions for the disturbances that we consider are that the velocity andtemperature perturbations are zero at the solid boundary at z = 0, and tend to zero awayfar from the wall, giving∂ψ∂x=∂ψ∂z= T = 0 on z = 0, and∂ψ∂x,∂ψ∂z, T → 0 as z →∞. (5)We look for solutions whereω(x, z, t) = ω(z)eλt cosαx, ψ(x, z, t) = ψ(z)eλt cosαx, T (x, z, t) = T (z)eλt sinαx, (6)where ω(z), ψ(z) and T (z) are real functions. The governing equations reduce tod2ωdz2=λωσ+ α2ω + αRaT, (7a)d2ψdz2= α2ψ − ω. (7b)d2Tdz2= λT + α2T − αψT ′ = λT + α2T + αψe−z2/4pi1/2, (7c)This system of ordinary differential equations can be solved to give the growth rate, λ, as aneigenvalue for any given Ra, σ and α using standard numerical techniques, or for marginalstability, λ = 0, we can find solutions with Ra as the eigenvalue. Results for σ = 7, a valueappropriate for water, are shown in figure 2. The lowest curve is for marginal stability, λ = 0,and is independent of the Prandtl number. The lowest point of this marginal stability curveis on the vertical axis, showing that the instability is a long-wave instability (although thereis no solution to the perturbation equations for α = 0). To find the point where the curveadjoins the axis we look for an asymptotic expansion in the limit of small α. We will look forthe leading order behaviour in the asymptotic expansion of Ra for marginal stability givenbyRa = Ra0 + αRa1 + α2Ra2 + · · · . (8)The solution is divided into two regions. The first, the inner layer, is near the wall withz = O(1), the length-scale of the temperature gradient. The second region, the outer layer,4FIG. 2: Stability boundary showing the critical value of Ra under the quasi-static approximationfor the error function temperature profile. Also show are growth rate contours with intervals of 0.01,with σ = 7. The dashed line shows the first two term of the asymptotic expansion Ra = Ra0+αRa1.is on the scale z = O(α−1), the scale of the along-wall wavelength of the disturbances. Inthe inner layer we pose the asymptotic expansionsω(z) = ω0(z) + αω1(z) + · · · , (9a)ψ(z) = ψ0(z) + αψ1(z) + · · · , (9b)T (z) = αθ0(z) + α2θ1(z) + · · · , (9c)In the outer layer we use the rescaled coordinate Z = αz, and pose the asymptotic expansionsω(Z) = Ω0(Z) + αΩ1(Z) + · · · , (10a)ψ(Z) = α−2Ψ0(Z) + α−1Ψ1(Z) + · · · , (10b)T (Z) = αΘ0(Z) + α2Θ1(Z) + · · · , (10c)The leading order equations in the inner layer areω′′0 = 0, (11a)ψ′′0 = −ω0, (11b)θ′′0 = ψ0e−z2/4/pi1/2, (11c)5while the leading order equations in the outer layer areΩ′′0 = Ω0 + Ra0Θ0, (12a)Ψ′′0 = Ψ0 − Ω0. (12b)Θ′′0 = Θ0, (12c)In the outer layer the background temperature gradient term is exponentially small and sois ignored. The boundary conditions imposed in the outer layer are that all variables decayas Z →∞. With this constraint the solutions areΘ0 = A0e−Z , (13a)Ω0 = B0e−Z − Ra0A02Ze−Z , (13b)Ψ0 = C0e−Z +B02Ze−Z − Ra0A08(Z2 + Z)e−Z , (13c)where A0, B0 and C0 are constants.The inner layer equations (11a) and (11b) have solutionsω0 = a0 + b0z, (14a)ψ0 = −a0z22− b0z36+ c0 + d0z, (14b)where a0, b0, c0 and d0 are constants. The no-slip boundary condition at z = 0 givesc0 = d0 = 0. We now match the streamfunction and vorticity between the layers by usingan intermediate variable or Van Dyke matching rules [3, 4]. This gives the requirement thatthe cubic term in ψ0 vanishes, as must the constant and Z terms in the small-Z Taylor seriesexpansion of Ψ0. Henceb0 = 0, C0 = 0,B02− RaA08= 0. (15)Matching the quadratic terms in the two expansions for the streamfunction gives−a02= −B02(= −Ra0A08). (16)Lastly, solving (11c) givesθ0 =2a0pi1/2(1− e−z2/4)+a0pi1/2∫ z0∫ ∞z′′e−z′2/4dz′ dz′′ + e0 + f0z, (17)6(a) (b) (c)FIG. 3: Contour plots of the composite asymptotic expansions for the leading order terms in theexpansions for the marginally stable mode of (a) vorticity, (b) streamfunction and (c) temperature,for α = 0.1.where e0 and f0 are constants. As θ0(0) = 0, so e0 = 0. Matching with Θ0 in the outer layergives f0 = 0 andlimz→∞θ0(z) =4a0pi1/2= A0. (18)Comparing (16) and (18) givesRa0 = pi1/2, (19)the critical value of the Rayleigh number in the limit α→ 0. This analysis can be extendedto higher orders. We find that the second term in the asymptotic expansion for Ra isRa1 = 3pi/2. The approximation given by these first two terms in this expansion is shownby the dashed line in figure 2, where we see it is a tangent to the stability boundary atα = 0, as would be expected.The form of the instabilities is shown in figure 3. These show the composite asymptoticexpansions [3, 4] of the leading order solutions of the vorticity, streamfunction and temper-ature which are valid in both the inner and outer layers. The solutions to the vorticity andthe streamfunctions are just the outer layer solutions which extend through the inner layer,while the temperature shows the adjustment to the zero boundary condition at the wall inthe narrow inner layer.7III. CONCLUSIONSWe have seen here that the critical Rayleigh number for a semi-infinite body of fluid withan error function temperature profile is Ra =√pi. This analysis uses the quasi-static, orfrozen-time, assumption and is based on a small-wavenumber asymptotic expansion. Thisresulting Rayleigh number is three orders of magnitude smaller than the critical value foundfor a layer with no-slip boundaries on the assumption the the boundary separation andthe temperature difference across the layer are the same. The reason for this is that theinstabilities in the layer are constrained to have a length-scale of the order of the layer height.This smaller scale greatly increases the effect of viscous damping.It should be noted that, of course, the quasi-static assumption is not strictly valid forthe semi-infinite body of fluid. However this same assumption is effectively being made incomparing the results of a layer whose height matches the instantaneous thickness of thethermal layer.[1] I. G. Currie, J. Fluid Mech. 29, 337 (1967).[2] P. G. Drazin and W. H. Reid, Hydrodynamic stability (Cambridge University Press, 1981).[3] M. Van Dyke, Perturbation Methods in Fluid Mechanics (Parabolic Press, 1975).[4] E. J. Hinch, Perturbation Methods (Cambridge University Press, 1991).8

Critical Rayleigh number of an error function temperature profile with a quasi-static assumption

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