Convective instability in a time-dependent buoyancy driven boundary layer

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Abstract

The stability of the parallel time-dependent boundary layer adjacent to a suddenly heated vertical wall is described. The flow is investigated through experiments in water, through direct numerical simulation and also through linear stability analysis. The full numerical simulation of the flow shows that small perturbations to the wall boundary conditions, that are also present in the experimental study, are responsible for triggering the instability. As a result, oscillatory behaviour in the boundary layer is observed well before the transition to a steady two-dimensional flow begins. The properties of the observed oscillations are compared with those predicted by a linear stability analysis of the unsteady boundary layer using a quasi-stationary assumption and also using non-stationary assumptions by the formulation of parabolized stability equations (PSE).

Introduction

An understanding of transient natural convection is of great importance in industrial applications where sudden changes in the surface boundary conditions can cause the heat transfer characteristics of a device to change significantly. Such situations arise for example in cooling systems and in crystal growth procedures. This paper focuses on the transient flow induced by the sudden heating of a vertical plate. Idealised in an infinite flow domain, heat transfer takes place purely by conduction and the one-dimensional form for the Navier–Stokes and energy equations is satisfied. In practice, finite geometry means that the symmetry is broken at the bottom of the heated element. The effect of this leading edge is propagated downstream producing a steady flow with a small vertical temperature gradient and a small horizontal velocity component is present. The analysis by Siegel [1] was the first to describe the behaviour of the leading edge effect in this way. Various experimental studies (summarized by Gebhart et al. [2]) have confirmed this basic flow regime. Other studies have attempted to predict the velocity at which the leading edge signal propagates up through the flow [3], [4], [5].

A study by Joshi and Gebhart [6] examined experimentally the breakdown of the one-dimensional flow using a constant heat flux boundary condition on a semi-infinite vertical wall. By making localised measurements of temperature and velocity, they found that the temperature and velocity traces deviated from the one-dimensional solution simultaneously at all downstream locations. Hence deviations from the one-dimensional flow occurred at some locations before the arrival of the leading edge signal. This breakdown of the one-dimensional flow suggests that it may become unstable before the leading edge signal arrives. However, this breakdown has not been observed previously in numerical studies of transient flow with isothermal or constant flux boundary conditions [5], [7], [8]. Since no instability has been observed in numerical simulations this indicates that any instability, where it exists, must be convective rather than absolute in nature [9]. Hence, any disturbances in the numerical simulation that occur in an unstable region of the flow may not be observed because they are carried away to a stable region of the flow before their amplitude is great enough to be observed. In the experiments the natural disturbance level may be higher and hence may allow the disturbances to be observed.

Considering the initial-value problem for a small disturbance super-imposed on a parallel flow, one can find the general solution for an initial disturbance that consists of a sinusoidal input with a constant frequency at some location in space. The solution is given by the homogeneous linear stability equation with zero temporal amplification, and the frequency is given by the input forcing. This problem is known as the spatial signalling problem [9]. It is also useful in flow situations where the flow is steady but weakly non-parallel where, by making certain approximations, a spatial growth rate and wavenumber that vary slowly in space can be found. An alternative problem for a steady parallel flow is the temporal signalling problem. In this case the forcing is for only an instant in time but over the whole flow domain in a sinusoidal function in space with a constant wavelength. The solution to this problem is given by the solution to the homogeneous stability equations with zero spatial amplification and the wavelength of the input forcing [10]. In a time-dependent parallel flow, where the temporal variation is slow, the temporal signalling problem may be extended by finding a temporal growth rate and frequency that vary slowly in time.

In considering the stability of the parallel flow adjacent to a suddenly heated wall Krane and Gebhart [11] compared the frequencies observed by Joshi and Gebhart [6] to the results of a quasi-stationary linear stability analysis. They found that the experimentally observed frequencies lay above the frequency which had the maximum amplification rate predicted from their stability analysis. The use of the quasi-static assumption was cited as the probable cause for this discrepancy.

Here we examine the flow of water adjacent to a vertical wall where the temperature of the wall is abruptly raised above that of its isothermal surroundings. The flow is examined experimentally, then using a full numerical simulation of the Navier–Stokes equations, and finally through linear stability analysis. The experimental setup is described in Section 2. A time series of temperature measurements at various locations within the boundary layer are taken with fast response thermistors. The numerical method is described in Section 3. Three different disturbance forcing modes are utilised in the flow simulations. Firstly a zero disturbance level is used which allows the simulation of the flow in the absence of convective instabilities. Perturbations are then introduced through continuous random heat sources throughout the boundary layer. This facilitates the comparison to the disturbance structures seen in experiments. Finally, a perturbation using a pulsed heat input at the start of the simulation with a sinusoidal structure in space is applied. This allows direct comparison to linear stability studies. The linear stability equations are described in Section 4. Stability equations using a quasi-stationary assumption are equivalent to the Orr–Sommerfeld equations (OSE) that arise by making parallel flow assumptions for a steady flow. The parabolized stability equations (PSE) are developed for a non-stationary parallel flow. These equations are analogous to the non-parallel parabolized stability equations used for stationary but spatially developing flows. The combined results are presented in Section 5 and discussed in Section 6.

Section snippets

Experimental setup

The experimental rig which models the semi-infinite plate was constructed from an existing facility previously used for experiments in a side-heated square cavity (for details of that cavity, see Patterson and Armfield [12]). Therefore, only a brief description is given here.

The cavity containing the working fluid is 24 cm wide by 31.5 cm high and 50 cm in the transverse dimension. One of the side walls of the cavity serves as the model for the semi-infinite plate. It consists of a 24×50 cm

Numerical analysis

The flow configuration is a square cavity of height H. The initial temperature, T within the cavity is uniform and equal to T0 and the vertical and horizontal velocities, u and v, are zero. The flow is initiated by raising the temperature of the left-hand-side wall to Th at time t=0. The right-hand-side wall is maintained at T0 and the top and bottom walls have adiabatic thermal conditions. The equations solved are the two-dimensional Navier–Stokes and energy equations with the Boussinesq

The basic flow

The transient behaviour of the fluid before it is affected by the leading edge can be modelled as if the plate were doubly infinite. In this case there is no vertical dependency in the flow and the horizontal velocity is zero. The continuity equation is trivially satisfied and the governing , , , are reduced to,∂u*∂t*2u*∂y∗2+gβ(T*−T0),∂T*∂t*2T*∂y∗2.For an isothermal wall boundary condition, these equations were first solved by Illingworth [16] and various solutions for uniform flux and

Results

The results are separated into those dealing with random disturbances and those with discrete wavelength disturbances. In Section 5.1, random perturbations are examined. In the experimental situation the random perturbations are introduced through unavoidable vibrations that occur through the startup process and through inhomogeneities in the boundary conditions. In the numerical simulation the perturbations are introduced through thermal perturbations at the hot wall. The introduction of

Discussion

The experimentally observed oscillations that occur before the arrival of the wavepacket are qualitatively similar to those in the numerical simulations. Although some discrepancy is observed between the signals in the furthest downstream location, the spatial inhomogeneity of the disturbance level in the experiment may result in the larger amplitude in the oscillations at this height. It is also noted that the disturbance amplitude on the steady flow is higher in the simulation than in the

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