Unsteady natural convection in a reservoir sidearm induced by time-varying isothermal surface heating

International Journal of Thermal Sciences 71 (2013) 61e73

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Unsteady natural convection in a reservoir sidearm induced by time-varying isothermal surface heating Yadan Mao a, *, Chengwang Lei b, John C. Patterson b a

State Key Laboratory of Geological Processes and Mineral Resources, Institute of Geophysics and Geomatics, University of Geosciences, Wuhan 430074, China b School of Civil Engineering, The University of Sydney, NSW 2006, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 August 2012 Received in revised form 13 January 2013 Accepted 15 March 2013 Available online 25 April 2013

Nearshore natural convection induced by differential heating across shore has significant biological and environmental implications. The present investigation is concerned with convection in a wedge-shaped reservoir sidearm induced by isothermal heating at the water surface which increases linearly over a specified time period from an initial value to a final steady value (i.e. ramped heating), extending our previous report on convection induced by constant isothermal heating. A semi-analytical approach coupled with scaling analysis and numerical simulation is adopted to resolve the problem. It is revealed that the flow domain is composed of two distinct subregions, a conductive region nearshore which finally becomes isothermal and stationary, and a convective region offshore which eventually maintains a steady flow velocity. Different scenarios are revealed through the comparison between the ramp duration P and a characteristic time tc of the induced flow. For the conductive region, the characteristic time is the time for the thermal boundary layer formed at the surface to reach the full local depth. For the convective region, it is the steady state time when convection balances conduction. For a short ramp (P < tc), no steady state is reached within the ramping stage. For a long ramp (P > tc), a quasi-steady state is reached for both subregions before the ramp finishes. For the conductive region, the quasi-steady state is characterized by a steady flow velocity and the same local temperature growth rate as the water surface. For the convective region, the quasi-steady state is characterized by a change in the growth rate of the velocity. Major time and velocity scales governing the flow development in both subregions are proposed by a semi-analytical solution coupled with scaling and verified by the numerical simulation. Implication of the present study for field situations is also discussed. Ó 2013 Elsevier Masson SAS. All rights reserved.

Keywords: Nearshore natural convection Buoyant boundary layers Time-varying isothermal heating Gravity currents Topographic effects

1. Introduction Field experiments [1e6] have revealed that natural convection in the nearshore regions of natural water bodies induced by differential heating across shore significantly reduced the residence time of water parcels, and thus mitigating the risk of eutrophication and pollution. This mechanism is often referred to as a “thermal siphon”. For nearshore regions under relatively calm conditions, this mechanism becomes prominent in refreshing the water bodies. In the experiments of Niemann et al. [5], the thermally-induced cross-shore currents were observed to be 2- to 5-fold higher than long-shore currents near the sea bed of Gulf Aqaba, Red Sea. Increasing recent interest has been paid to the biological implications of this cross-shore circulation [3e7]. Apart from

* Corresponding author. Tel./fax: þ86 27 67883251. E-mail addresses: [email protected], [email protected] (Y. Mao). 1290-0729/$ e see front matter Ó 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2013.03.013

promoting exchange between nearshore and open waters in reservoirs and lakes [3,4], e.g. the transport of eutrophication chemicals such as phosphorus, field experiments of Niemann et al. [5] and Monismith et al. [6] suggest that the thermal siphon may be a general feature of the hydrodynamic processes of reefs and the coastal ocean, helping to alleviate the stress of coral bleaching [8,9] as well as enhancing connectivity between the reef and the ocean (e.g. the transport of near-reef phytoplankton to deeper regions by gravity currents). The biological and environmental significance of the thermal siphon revealed by field studies has motivated an increasing effort in understanding the flow mechanisms and the quantification of flow properties through a variety of complementary approaches for different thermal forcing conditions. In a broader context, research of horizontal convection has been framed in a rectangular domain with non-uniform thermal forcing imposed on the horizontal boundary, which directly initiates a horizontal temperature gradient [10e14]. A brief review of the

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Y. Mao et al. / International Journal of Thermal Sciences 71 (2013) 61e73

work in this area can be found in Mao et al. [15]. The rectangular model neglects an important mechanism, that is, the differential heating or cooling imposed intrinsically by the depth variation across shore. When the water surface is exposed to the same amount of heating or cooling, the depth-averaged temperature decreases or increases in the offshore direction as the water depth increases, generating a horizontal temperature gradient that drives the flow. A wedge model that we adopt in this study takes this bathymetry effect into account. Unlike the above-mentioned investigations which adopt a non-uniform thermal forcing and a rectangular domain, the horizontal convection in the present investigation results from uniform thermal forcing at the water surface of a wedge domain. In the present model, differential heating is induced by the varying bathymetry across shore, and the key in generating the flow is the depth variation which results in the horizontal temperature gradient. To capture the above-described feature of varying bathymetry in the nearshore region, a wedge model has been adopted in many investigations. For constant radiation heating, which results in a non-uniform heat source distribution over the local water depth, theoretical quantification of the thermal flow has been achieved through zero-order asymptotic solutions [16] and scaling analysis [17,18]. The stability properties of this configuration have been investigated by linear stability analysis [19], direct numerical simulations [20] and spectral analysis [21]. The problem of constant cooling by an imposed heat flux at the water surface has been investigated through numerical simulations and scaling analysis [22e27]. The problem of alternate heating and cooling over diurnal cycles has also been investigated in a number of studies: The asymptotic solutions of Farrow and Patterson [28] and Farrow [29] assume that a periodic surface heat flux is uniformly distributed as internal sources and sinks over the local depth, and the numerical simulation of Lei and Patterson [30] assumes an exponential decay (absorption) of the radiative flux over the local depth during the heating phase, and a surface heat flux during the cooling phase. Almost all of the existing theoretical investigations on this topic with a wedge model have adopted a constant heat flux at the water surface, or an internal heating source, or both. Among the many different modes of heat transfer between the atmosphere and water body (sensible heat transfer, radiative heat flux, latent heat flux etc.), here we focus on natural convection induced by sensible heat transfer alone to evaluate its role in the overall thermal convection. For the sensible heat transfer, i.e. heating or cooling induced by a temperature difference between the water body and the atmosphere, the thermal forcing may be represented by a water surface with a temperature difference from that of the inner water body. To this end, the laboratory experiments of Bednarz et al. [31,32] study the convective flows under an isothermal surface with both constant and sinusoidally varying temperatures. More recently, our theoretical investigation of constant isothermal surface heating [15] has identified different flow subregions and provided detailed scaling laws for each subregion. Since in field situations the atmospheric temperature usually varies over time, as a further step toward the field situations, the present study extends our previous investigation of constant heating [15] by considering a more complex case with ramped isothermal surface heating, in which the surface temperature increases linearly with time over a certain time duration P, and then remains constant. This is again a simplified case of the field situations, and the purpose is to provide insight into the timedependent dynamics of the induced flow. A comparison of the ramp duration with other time scales of the flow development may result in different scenarios, providing insights into the effect of the variation of thermal forcing on the buoyancy-driven flow. Results of the present study will be compared with that of the constant

isothermal heating case [15] to highlight their differences. Focus will be on the flow development over the ramping duration P when the thermal forcing varies with time. The linearly increasing temperature can approximately represent the temperature variation from early morning to mid-afternoon or weekly average temperature from spring to summer, depending on the duration of the ramping. With further addition of a ramped cooling phase, this can ultimately lead to investigations of periodic thermal forcing with consecutive ramped heating and cooling representing the diurnal or seasonal variation. In general, the linearly increasing temperature represents the simplest form of unsteady temperature change, and provides a basic case for studying the flow response relative to varying thermal forcing which can easily be extended to more complex time-dependent thermal forcing conditions. 2. Model formulation The geometry of the present model is the same as that considered in Mao et al. [15], but the form of the thermal forcing is different between the two models. Here we briefly describe the model for the sake of completeness of the content, while more details of the configuration and the reasoning behind it can be found in Mao et al. [15]. We consider a cross-shore region of a reservoir consisting of one section bounded by a sloping bottom with a slope A and the other section with uniform depth (see Fig. 1). With the Boussinesq assumption, the two-dimensional Naviere Stokes and energy equations governing the flow and temperature evolution are written as:

vu vv þ ¼ 0; vx vy

(1)

vu vu vu 1 vp þu þv ¼  þ nV2 u; r0 vx vt vx vy

(2)

vv vv vv 1 vp þu þv ¼  þ nV2 v þ g bðT  T0 Þ; r0 vy vt vx vy

(3)

vT vT vT þu þv ¼ kV2 T: vt vx vy

(4)

where u and v are the horizontal and vertical velocity components, respectively; T is the fluid temperature and p is the pressure. The parameters r0, k, n and b are respectively the density, thermal diffusivity, kinematic viscosity and thermal expansion coefficient of the fluid at a reference temperature T0. In the present model, the flow is initially stationary and b ¼ 0, where isothermal at a temperature of T0. An adiabatic (vT=v n b is the outward normal unit vector to the surface) and rigid no-slip n (u ¼ 0, v ¼ 0) boundary condition is assumed for the bottom, including both the sloping and the flat sections, whereas the water surface is assumed stress free (vu=vy ¼ 0, v ¼ 0). An open boundary condition is considered for the endwall (vu=vx ¼ 0, v ¼ 0,

Fig. 1. Sketch of the flow domain and the coordinate system.

Y. Mao et al. / International Journal of Thermal Sciences 71 (2013) 61e73

vT=vx ¼ 0), assuming that any backflow is at the initial (reference) temperature T0. The temperature at the water surface (Ty¼0) first increases linearly with time over a certain time duration of P and then becomes constant, as expressed below:

 Ty¼0 ¼

T0 þ ðDT=PÞt T0 þ DT

0tP t>P

(5)

Results obtained under this time-varying thermal forcing will be compared with those obtained under the constant isothermal heating to highlight the effect of the variation of thermal forcing with time on flow development, especially the effect of the ramping duration on the flow scenario, shedding light on the nearshore flow development induced by sensible heat transfer during the heating phase of daily or seasonal cycles. The results may also have implications for varying thermal forcing imposed on other configurations. 3. Numerical procedures The governing equations (1)e(4) along with the specified boundary and initial conditions are solved numerically using a finite-volume method. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) scheme is adopted for pressureevelocity coupling; and the QUICK (Quadratic Upstream Interpolation for Convective Kinematics) scheme is applied for spatial derivatives [33]. A second-order implicit scheme is applied for time discretization in calculating the transient flow. To generalize the results of this study, all simulation results are presented in a normalized way. The respective scales for the normalization are: the length scale x, y wh; the time scale t, P wh2/ k; the velocity scale u, v wk/h; and the pressure gradient scale px, py wrgbDT. The non-dimensional temperature s is defined as s ¼ (T  Ty¼0)/DT, and thus the normalized temperature represents the temperature difference between the local and the water surface temperatures, with values ranging from 1 to 0. Accordingly, within the ramp duration P, a steady or quasi-steady state is reached if this temperature difference s becomes constant with time, i.e. the internal temperature increases at the same rate as the water surface. In addition to the bottom slope A, the following two non-dimensional parameters characterize the natural convection flow: the Prandtl (Pr) and Rayleigh (Ra) numbers, which are defined as

n k

Pr ¼ ;

Ra ¼

g bDTh3

nk

(6)

It is worth mentioning that hereafter, all the mathematical expressions are in non-dimensional forms. The simulation is conducted in a sidearm with a maximum dimensionless depth of y ¼ 1, two bottom slopes of A ¼ 0.05 and 0.1 respectively, and a fixed Prandtl number of Pr ¼ 7. The section of interest with a uniform depth is set to be of the same length as the sloping section. The condition that the backflow is at the initial (reference) temperature T0 results in some distortion of the temperature and flow fields near the far endwall. As this is not of relevance to the basic flow mechanisms under investigation, the computational domain is extended beyond the region of interest to minimise this effect. Based on the mesh and time-step dependency tests reported in Ref. [15], a mesh of 451  75 is selected for the region of interest and a dimensionless time-step of 2.0  104 is adopted for the simulation. The flow domain is meshed with a non-uniform grid with an increasing grid density toward all of the boundaries. For the chosen mesh, the minimum face area is 0.0044; the maximum face area is 0.8730; and the maximum stretch factor is 1.08.

63

4. Theoretical analysis and numerical simulations It is expected that a comparison between the ramp duration P and other characteristic time scales pertinent to the flow will reveal different scenarios of the flow development. In the constant isothermal heating case (that is P ¼ 0), our previous study [15] revealed two distinct flow subregions, i.e. a conductive region nearshore and a convective region offshore. The major differences between these two subregions in terms of both transient and steady state properties are illustrated in Fig. 2 which is a recalculation of the results of Ref. [15] using the extended computational domain. Fig. 2(a)e(d) shows respectively the isotherms and streamlines at steady state. The dashed lines shown in these figures approximately divide the entire domain into two subregions. It is revealed that the nearshore conductive region, marked as ‘I’, is isothermal and stagnant at the steady state. In contrast, the offshore convective region, marked as “II”, maintains a non-zero flow and a distinct thermal boundary layer at the steady state. The convective region expands toward the shore with increasing Ra as illustrated by comparing Fig. 2(a, b) with (c, d). Furthermore, the evolution of velocity and temperature in these two subregions for Ra ¼ 2  106 are plotted in Fig. 2(e) and (f) respectively. At x ¼ 1, representative of the conductive region, the flow velocity increases until the thermal boundary reaches the local bottom and eventually becomes zero [15]. The absolute value of temperature difference s decreases from one to zero as time goes on, indicating that the region eventually reaches the same temperature as the water surface. At x ¼ 3, representative of the convective region, a steady nonzero flow velocity and temperature difference is maintained at the steady state. As the thermal forcing in the present ramped heating case eventually becomes constant with time, a scenario similar to the constant isothermal heating case reported in Mao et al. [15] is expected for the final steady state of the flow, i.e. a conductive region nearshore and a convective region offshore. For the conductive region which is nearshore and shallow, heat convected away from the thermal boundary layer is insufficient to balance the heat conducted into it, and therefore the thermal boundary layer keeps growing until it reaches the local depth. For the convective region which is further offshore and deeper, as the convective flow velocity increases with time, at a certain stage convection becomes sufficiently strong to balance conduction and therefore the thermal boundary layer stops growing and remains distinct. However, during the ramping stage when the surface temperature increases linearly with time, the flow scenario is expected to be more complex than the constant heating case described above. In the ramping stage, the characteristic time scales of the flow development should be compared with the ramping duration, which results in short ramp and long ramp scenarios depending on the comparison. Under the ramped heating condition, the characteristic time scale for the conductive region, that is, the time for the thermal boundary layer to reach the local water depth, is expected to be the same as that under constant isothermal heating, since the growth rate of the thermal boundary layer from the surface is independent of the thermal forcing. However, this time scale depends on the longitudinal position within the sloping bottom region. After the ramp heating finishes, the flow is subject to constant isothermal heating, and therefore the conductive subregion is expected to eventually become isothermal and stationary, as in the constant isothermal heating case [15]. For the convective subregion, the characteristic time scale is the time at which a balance is achieved between the heat convected away from the thermal boundary layer and the heat conducted into it through the surface. A comparison of the characteristic time with the ramp duration P is expected to reveal two different scenarios. Scaling for the dividing position

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Y. Mao et al. / International Journal of Thermal Sciences 71 (2013) 61e73

Fig. 2. Simulation results for the constant isothermal heating (P ¼ 0) case: isotherms at steady state with an interval of 0.2 for (a) Ra ¼ 2  106 and (c) Ra ¼ 107. Streamlines at steady state with an interval of 10 for (b) Ra ¼ 2  106 and (d) Ra ¼ 107. (e) Time series of the maximum horizontal velocity along the vertical lines of x ¼ 1 within the conductive region, and of x ¼ 3 within the convective region for Ra ¼ 2  106. (f) Time series of the average temperature along the vertical lines of x ¼ 1 and x ¼ 3 for Ra ¼ 2  106.

between these two subregions has been predicted and verified in Mao et al. [15]. In what follows, scaling analysis is described separately for these two different subregions. It is worth noting that the transition region between the two subregions cannot be described by the scaling analysis, which is based on an identification of the dominant mechanisms in each subregion. In the transition region, both mechanisms are relevant and scaling analysis is not possible at this point of time. Consequently it is not expected that the scaling results for the two subregions will merge smoothly at the dividing position. 4.1. Conductive region

vs ¼ 0 vy

(7)

0  t < PÞ;

(8)

ðy ¼ AxÞ;

(9)

s ¼ 0 ðt ¼ 0Þ:

(10)

The above equations are solved by separation of variables, which gives the following solution:

s ¼

For t  P, the temperature at the water surface increases linearly with time. Similarly to the constant isothermal heating case, it is expected that conduction dominates over convection in the conductive region nearshore. The small slope assumption suggests that the thermal boundary layer thickness is much smaller than any horizontal length scale, which results in vertical derivatives much larger than horizontal derivatives, i.e. vertical conduction dominates over horizontal conduction. Therefore the problem is simplified as a one-dimensional conduction problem over a local water depth of Ax as described below in a non-dimensional form:

vs 1 v2 s ; þ ¼ vt P vy2

s ¼ 0 ðy ¼ 0;

N X

16A2 x2

1  exp þ 1Þ3 p3 P    1 y p  sin nþ : 2 Ax

n ¼ 0 ð2n

ðn þ 1=2Þ2 p2 t  A2 x2

!!

(11)

At the start-up (t ¼ 0), the exponential term is unity and hence the summation term is zero. Therefore, the normalized temperature difference s is initially zero as expected. As time increases, the exponential term decreases, and the summation term becomes more negative, indicating an increasing difference between the local and the water surface temperatures. The exponential term in Eq. (11) approaches zero as time becomes large. Therefore, the temperature difference becomes constant with time if the ramp period P is sufficiently large, and may be approximated by:

Y. Mao et al. / International Journal of Thermal Sciences 71 (2013) 61e73

ss ¼

N X

16A2 x2

n ¼ 0 ð2n

þ 1Þ3 p3 P

sin

   1 y p : nþ 2 Ax

65

(12)

It is seen from Eq. (11) that the summation term decays rapidly with n, and hence n ¼ 0 is the dominant term. Consequently, a quasi-steady state is reached at a time scale associated with the decay of the dominant exponential term in Eq. (11), which is:

tsp w

4A2 x2

p2

wA2 x2 :

(13)

The above dimensionless tsp is always less than 1 except at the maximum depth position (x ¼ 1/A) where it reaches the maximum value of 1. For t > tsp, the exponential term is asymptotically negligible and the flow becomes steady. It is worth noting that the steady state time scale (13) is of the same order as the diffusion time scale for the thermal boundary layer to reach the local bottom, which is A2x2. Therefore, for a given offshore distance x, this suggests two possible scenarios: (a) The long ramp scenario of P > tsp (i.e. the ramp period is sufficiently long): after the thermal boundary layer reaches the local bottom at a depth of Ax, the temperature difference s between the local position and the water surface becomes constant with a value specified by Eq. (12). In this case, a quasisteady state is reached during the ramping stage. (b) The short ramp scenario of P < tsp (i.e. the ramp finishes before the thermal boundary layer reaches the local bottom): a quasisteady state will not be reached during the ramping stage. From the quasi-steady state time scale (13), it is clear that for any finite ramp time P, there will be always a nearshore region where x is sufficiently small so that tsp < P and a quasi-steady state is reached within the ramp duration, and an offshore region with tsp > P (provided P < 1), where no quasi-steady state can be reached within the ramp duration. From Eq. (13), the extent of the nearshore region which satisfies the long ramp condition P > tsp is given by x < O(P1/2/A), where P1/2 represents the length scale of thermal diffusion over the ramping period. Therefore, the nearshore region extends out to the point where diffusion can just penetrate the full depth during the ramping period. This region where a quasi-steady state is reached within the ramp duration expands offshore as P increases. With a sufficiently small Ra, the entire domain may be conduction-dominated. In this case, since the non-dimensional length of the domain is 1/A, if O(P1/2/A) > 1/A, i.e. P > O(1), the entire domain will reach quasi-steady state during the ramping stage. To verify the analytical solutions, the time series of the temperature s at two x locations and two y values are extracted from the numerical simulations and are plotted in Fig. 3 along with the analytical solution (11) for the case A ¼ 0.1, for typical long (P ¼ 0.4) and short (P ¼ 0.04) ramps. It is worth noting that the analytical solution applies only to the ramping stage t < P. Two distinct scenarios of P < tsp and P > tsp are plotted in Fig. 3(a) and (b) respectively. It is clear that the variation of temperature with time is different for these two scenarios. The analytical solution in Eq. (11) suggests that for t < P, s should decrease until either t ¼ P (for a short ramp) or a quasi-steady state is achieved some time before P (for a long ramp). For the short ramp scenario of P < tsp, as shown in Fig. 3(a), the analytical and numerical solutions coincide up until the ramp finishes at t ¼ 0.04. This supports the assumptions made in the derivation of the analytical solution. In this case the ramp finishes before the thermal boundary layer reaches the local bottom at these specified x locations, and hence a quasi-steady state is not

Fig. 3. Time series of temperature from numerical simulations (solid and dashed lines) and analytical solutions (circles, applies for t  P) for A ¼ 0.1 (a) P ¼ 0.04 for the short ramp scenario of P < tsp. (b) P ¼ 0.4 for the long ramp scenario of P > tsp.

reached during the ramping stage at the selected positions. However, for the very nearshore region x < P1/2/A where tsp < P, a quasisteady state will be reached, as will be illustrated later in Fig. 5b. After the ramp finishes at t ¼ 0.04, the numerical results show that the temperature difference s gradually approaches zero, which suggests that the local water eventually reaches the same temperature as the water surface, as expected.

Fig. 4. Verification of the quasi-steady state time and temperature for the long ramp period of P ¼ 0.4. (a) Time series of temperature at different positions for different bottom slopes. (b) Time series normalized by the steady state scales.

66

Y. Mao et al. / International Journal of Thermal Sciences 71 (2013) 61e73

and of different bottom slopes. After normalizing the time and temperature with their respective scales, Eqs. (13) and (12) respectively, it is found that for t < P, all the time series of the normalized temperature collapse onto a single curve in Fig. 4(b), confirming the correctness of the steady state scaling of tsp and ss. From Eq. (11), the average temperature over the local depth for a given offshore distance x can be derived as:

s¼

N X

32A2 x2

n ¼ 0 ð2n

þ 1Þ4 p4 P

exp



ðn þ 1=2Þ2 p2 t A2 x2

!

! 1 :

(14)

Similarly, for the long ramp scenario of P > tsp, the exponential term in Eq. (14) decays with time and finally becomes negligible when t > tsp, and thus the temperature difference s between the vertically averaged temperature and the surface temperature becomes constant with a value of:

ss ¼ 

N X

32A2 x2

n ¼ 0 ð2n

þ 1Þ4 p4 P

:

(15)

The non-dimensional horizontal gradient of the vertically averaged temperature can be obtained from the derivative of Eq. (14):

! N X vs 64A2 x 16t þ ¼ 4 4 vx ð2n þ 1Þ2 p2 Px n ¼ 0 ð2n þ 1Þ p P ! ðn þ 1=2Þ2 p2 64A2 x : exp  t  A 2 x2 ð2n þ 1Þ4 p4 P

(16)

Since the summation terms in Eq. (16) decrease rapidly with n, the first term of n ¼ 0 dominates. Approximating the infinite summation in Eq. (16) with the first term, we have

    p2 vs 64A2 x 16t 64A2 x exp  t  þ : w vx p4 P p2 Px p4 P 4A2 x2

Fig. 5. Verification of the quasi-steady state time and flow velocity for the long ramp scenario of P > tsp. Time series of the maximum flow velocity over local depths for (a) P ¼ 0.4, (b) P ¼ 0.04. (c) Time series in (a) and (b) normalized by the respective quasisteady state time and velocity scales. For the same line style in (a) and (b), the value of x increases from bottom up.

For the long ramp scenario with P > tsp, as shown in Fig. 3(b), the ramp finishes after the thermal boundary layer reaches the local bottom, and as predicted by the analytical solution, a steady temperature difference s is observed in Fig. 3(b) up until the ramp finishes at t ¼ 0.4, indicating that, after the thermal boundary layer reaches the bottom, the local temperature increases at the same rate as the water surface during the ramping stage. The solution of Eq. (11) also agrees well with the results from the numerical simulations for t  P. After the ramp finishes at t ¼ P, the flow is under constant heating, and the simulation results show that the temperature difference between the local positions and the water surface gradually diminishes, suggesting that the flow finally becomes isothermal again. It is worth noting that the values of s in Fig. 3 are significantly smaller for longer ramping periods P, which is consistent with the 1/P dependence in Eq. (11). Therefore, during the ramping stage, for long ramping periods the temperature of the conductive region follows more closely the surface temperature than for short ramping periods. For the long ramp scenario where a quasi-steady state occurs within the ramp duration, to verify the steady state scales, Fig. 4(a) plots the numerical results of time series of s at different positions

(17)

Similarly, for the long ramp scenario of P > tsp, the exponential term in Eq. (17) decays with time and finally becomes negligible when t > tsp, and thus the horizontal temperature gradient becomes steady at a scale of:

 vs 64A2 x w 4 :  vx s p P

(18)

The temperature gradient specified in Eq. (17) generates a pressure gradient that drives the flow. Following the same procedure described in Mao et al. [15] for the case of constant isothermal heating, a balance between buoyancy induced pressure gradient and viscous terms in the momentum equations yields the following scaling for velocity:

   2 4  2 4  p2 16A3 4A x 4A x 2 exp  uw Ra  þ tx t : p2 p2 P p2 4A2 x2

(19)

Similar to the temperature result, there are two scenarios for velocity. Details are given below. 4.1.1. The long ramp scenario For the long ramp scenario of P > tsp, the exponential term in Eq. (19) decays with time and becomes negligible after the start of quasi-steady state, i.e. t > tsp, and thus the flow velocity becomes steady within the ramping stage with a scale of

usp w

64A5 Rax4 A5 Rax4 w : P P p4

(20)

Y. Mao et al. / International Journal of Thermal Sciences 71 (2013) 61e73

The above analysis suggests that for the long ramp of P > tsp, at time tsp, the thermal boundary layer reaches the local bottom and a quasi-steady state is reached locally with a flow velocity described by Eq. (20). Time series of flow velocity are obtained from numerical simulations and plotted in Fig. 5(a) and (b) for two ramp durations P ¼ 0.4 and 0.04 respectively. For both ramping durations, a nearshore region which satisfies the long ramp criterion of P > tsp can be identified. For the relatively short ramping periods of P ¼ 0.04, the selected positions are very close to shore, since tsp increases with x and therefore the criterion of P > tsp is not satisfied at larger x, as is shown in Fig. 3(a) where a quasi-steady state is not reached at the selected relatively large offshore distances for P ¼ 0.04. Therefore, the duration of the ramping stage is deemed short or long only with respect to tsp for the positions under consideration. As mentioned before, the region where a quasi-steady state is reached is represented by x < O(P1/2/A). Therefore, for the same A, the selected offshore distances for P ¼ 0.04 are smaller than those selected for P ¼ 0.4, and for the same P, the selected offshore distances are smaller for A ¼ 0.1 than those for A ¼ 0.05. For both ramping durations, where P > tsp is satisfied, a quasi-steady state can be observed in Fig. 5(a) and (b) before the ramp finishes. After the ramp finishes, as the region gradually becomes isothermal with the water surface temperature, the driving force gradually diminishes. As a consequence, the velocity decreases and eventually approaches zero. Fig. 5(c) re-plots the time histories of the velocity shown in Fig. 5(a) and (b) after normalization by the steady state time scale (13) and the corresponding velocity scale (20). It is seen in this figure that the early and steady stages of the time series of velocity at different parametric settings collapse onto a single curve, verifying the scaling prediction of the steady state time and velocity. Compared to Fig. 5, more steady state velocity data extracted from the numerical results are plotted in Fig. 6 against the scaling prediction. The clear linear correlation in Fig. 6 again verifies the scaling prediction of the velocity dependency on A, Ra, P and the offshore distance.

 uw4RaAtexp



p2 4A2 x

 t : 2

67

(21)

Letting the derivative of u with respect to t be zero reveals that the maximum velocity occurs at a time scale of

tm w

4A2 x2

p2

wA2 x2 ;

(22)

which is the same as the time scale for the thermal boundary layer to reach the local depth. The scale for the maximum velocity is obtained by substituting Eq. (22) into Eq. (21) as:

umax wRaA3 x2 :

(23)

It is expected that these scales for the constant heating case also apply to the case of short ramp periods of P < tsp, since the flow development after the ramping period is under constant heating. In other words, after time P, the flow develops in the same manner as the constant heating case. To verify this, time series of the maximum velocity over local depth at different offshore distances and for different Rayleigh numbers are plotted in Fig. 7(a) for P ¼ 0.04. This ramp period is too short for the thermal boundary layer to reach the local water depth at the selected locations. It is seen that after the ramp finishes at P, the flow velocity continues to increase with time until it reaches a maximum value and then decreases. In other words, the flow develops as if the ramp does not exist. After being normalized with the respective scales of Eqs. (23) and (22), which represent the maximum velocity and the corresponding time for it to occur, all the maxima in the time series in Fig. 7(a) collapse onto a single point in Fig. 7(b), confirming that the scales of the maximum velocity and the corresponding time derived for the constant heating case [15] are also applicable to the short ramp heating case of P < tsp.

4.1.2. The short ramp scenario For the short ramp scenario of P < tsp, the ramp finishes before the thermal boundary layer reaches the local depth. After the ramp finishes, the temperature at the water surface becomes constant, and therefore, the subsequent flow development will be under constant heating. For the constant heating case, scales for the conductive region have been derived for the maximum flow velocity in the time series and the corresponding time by Mao et al. [15]. After normalization, the unsteady velocity scale is written as:

Fig. 6. The steady state velocity within the ramp duration from numerical simulations versus scaling predictions.

Fig. 7. Verification of the maximum velocity and the corresponding time for the short ramp scenario with A ¼ 0.05 and P ¼ 0.04. (a) Time series of the maximum flow velocity at different offshore distances for different Ra. (b) Time series normalized by the scales of the maximum velocity and the corresponding time respectively.

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The above discussion concentrates on the conductiondominated region for both the long and short ramp scenarios. For the long ramp scenario, a quasi-steady state is reached within the ramping stage and the relevant quasi-steady state scales are proposed and verified. For the short ramp case, no quasi-steady state is reached within the ramping stage, and the scales for the maximum velocity and the time for its occurrence are the same as those for the constant heating case. Numerical results show that for both the long and short ramp scenarios, the flow in the conductive region gradually becomes isothermal and stationary after the ramp finishes, which is the same as the constant heating case [15]. 4.2. Convective region The above section describes the flow development in the conductive region where the flow eventually becomes isothermal and stationary. As demonstrated above, the development of the flow in the conductive region is well quantified by a hybrid of analytical solution and scaling analysis. However, if convection is significant, the analytical solution will no longer hold. In the offshore region, if convection is sufficiently strong to balance conduction, the thermal boundary layer will stop growing and remain distinct, and a steady flow velocity is expected to be maintained. The characteristic time scale for the offshore convective region is the time when convection balances conduction. Comparing the characteristic time with the ramping time P results in two possible scenarios: (i) The short ramp scenario in which no steady state is reached within the ramp duration: In this case, over the ramping duration of 0 < t < P, the flow is controlled by ramp heating and no quasi-steady state is reached. As the ramp heating finishes (t > P), the flow is controlled by constant heating and eventually becomes steady. (ii) The long ramp scenario in which a quasi-steady state is reached within the ramp duration: In this case, the flow becomes quasi-steady when convection balances conduction. This happens during the ramping stage. Subsequently the flow is still driven by ramp heating and remains quasi-steady within the ramp stage. As the ramp duration ends (t > P), the flow is subject to constant heating and becomes steady. The following scaling analysis aims at deriving scales for flow properties at different stages and the corresponding time scales for the above two scenarios respectively. At the initial stage, and for t < P, the flow velocity is small and thus convection is negligible. In the vertical momentum equation (3), a balance between the pressure gradient term and the buoyancy term provides the scale for the non-dimensional pressure p:

pwdT t=P:

(24)

A balance between the unsteady term and the diffusion term in the energy equation yields the scale for the thickness of the thermal boundary layer

dT wt

1=2

:

(25)

In the horizontal momentum equation (2), comparison of the unsteady inertia term O(u/t), the advection term O(u2/x), and the 2 viscous term Oðnu=dT Þ reveals that the viscous term dominates among the three terms at small times if Pr > 1. Therefore, in the horizontal momentum equation (2), a balance between the viscous term and the pressure gradient yields a scale for velocity within the ramping duration:

3

RadT t uw : xP

(26)

Substituting Eq. (25) into Eq. (26), the velocity scale at the initial stage is derived as

uwRa

t 5=2 : xP

(27)

Scale (27) indicates that the flow velocity increases with time, and so does the convection. If convection is strong enough to balance conduction, then the flow will become steady or quasi-steady and the thermal boundary layer will remain distinct. A balance between the conduction term and the convection term in the energy equation is written in a non-dimensional form as:

us s w : x d2

(28)

T

4.2.1. The short ramp scenario For the short ramp scenario, the thermal boundary layer will not reach a steady state during the ramping stage. Instead, it will become steady after the ramp finishes at P. Since after time P, the flow is under constant heating, and therefore the velocity scale at the steady state is expected to be the same as that for constant heating, similar to the case of a non-instantaneously heated vertical wall described in Patterson et al. [34]. For the constant isothermal heating case, previously we have derived and verified the scale for the steady state time, as well as steady state scale for the flow velocity and the thickness of the thermal boundary layer as follows [15]:

tsc wRa2=5 x4=5 :

(29)

usc wRa2=5 x1=5 ;

(30)

dTc wx2=4 Ra1=5 :

(31)

A distinct thermal boundary layer must be maintained for the convective region. Therefore, it should be satisfied that dT < Ax, that is

x > A5=3 Ra1=3 ;

(32)

which suggests that the convective region expands toward the shore as Rayleigh number or the bottom slope increases. For the short ramp scenario, the velocity scale at the end of the ramp heating can be derived by simply replacing t with P in Eq. (27):

up wRaP 3=2 =x:

(33)

To verify the scales for the short ramp scenario, the time series of the maximum horizontal velocity along x ¼ 3.5 are plotted in Fig. 8(a) for different Rayleigh numbers. Fig. 8(b) plots the same set of data with the time and velocity normalized by P and up in Eq. (33) respectively. It is seen in Fig. 8(b) that the early parts of the four different curves (up to the end of the ramp) collapse together, suggesting that the velocity scale at P for the short ramp case is well predicted by Eq. (33). For the verification of the steady state scales,

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in Fig. 9(a) for different Ra and P. Fig. 9(b) re-plots the same set of data with the time and velocity normalized by Eqs. (34) and (35) respectively. It is clear in Fig. 9(b) that different curves collapse together at the initial and the quasi-steady stages, suggesting that the dependency of the quasi-steady state scales on Ra and P is well quantified. Another feature revealed by Fig. 9(a) is that for the same Rayleigh number, as soon as the ramp finishes, velocities under different ramp durations P reach the same steady state value, irrespective of the ramp duration (as long as it falls into the long ramp scenario of P > ts). The scale for this steady state value is derived and verified below. Since a prerequisite for the existence of a quasi-steady state within the ramp duration is ts < P, using the scale of Eq. (34), the condition is derived as:

x < P 5=4 Ra1=2 :

(36)

In addition, for convective region, the thickness of the thermal boundary layer at the quasi-steady state should be smaller than the < Ax or: local depth, i.e. t1/2 s

x > P 1=5 Ra1=5 A7=5 :

(37)

Therefore, the spatial extent for the quasi-steady state to occur in the convective region is

P 1=5 Ra1=5 A7=5 < x < P 5=4 Ra1=2

(38)

The criterion for the existence of a quasi-steady state in the convective region can be derived by comparing the left and the right side of Eq. (38) as

Ra > P 3=2 A2

(39) 3/2 2

If condition (39) is not satisfied, i.e. Ra < P A , no quasisteady state is reached within the ramping duration for the convective region. In this case, the steady state in the convective

Fig. 8. Time series of horizontal velocity at x ¼ 3.5 in the convective region for the short ramp scenario, P ¼ 0.04 < ts. (a) Numerical results for different Rayleigh numbers. (b) Numerical results normalized by scaling in Eq. (33) for velocity at time P. (c) Numerical results normalized by the steady state scales (29) and (30).

Fig. 8(c) shows the normalized results of Fig. 8(a) by the steady state time and velocity scales of Eqs. (29) and (30) respectively. The convergence of different time series at steady state confirms that the steady state scales for the constant heating case are also applicable for the short ramp scenario. 4.2.2. The long ramp scenario For the long ramp scenario, i.e. ts < P, a quasi-steady state is expected to occur within the ramp duration. Using Eqs. (25), (27) and (28), the scale for quasi-steady state time can be derived as:

ts wx4=7 P 2=7 Ra2=7 :

(34)

Thus, for the long ramp scenario, the time it takes to reach the quasi-steady state decreases with Ra and increases with P. The velocity at the start of the quasi-steady stage can be derived by substituting Eq. (34) into Eq. (27).

uss wx3=7 Ra2=7 P 2=7 :

(35)

To verify the above scales for the quasi-steady state, the time series of the maximum velocity along the line of x ¼ 3.17 are shown

Fig. 9. Time series of the maximum velocity along the vertical line of x ¼ 3.17 for the long ramp scenario of P > ts. (a) Numerical results for different Ra and P. (b) Numerical results normalized by the quasi-steady state scales of Eqs. (34) and (35).

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region is only reached under constant heating after the ramp finishes. In the convective region, for the long ramp scenario of P > ts, after the flow reaches the quasi-steady state, i.e. t > ts, convection balances conduction and thus the flow remains quasi-steady afterwards. In this case, the thermal boundary layer will not continue to grow at the scale of Eq. (25). Instead, it will adjust itself to maintain the balance between conduction and convection. Therefore, substituting Eq. (26) into Eq. (28), the thickness of the thermal boundary layer at the quasi-steady state can be derived as:



dT w

x2 P tRa

1=5 :

(40)

Substituting Eq. (40) into Eq. (26) yields the velocity scale of the quasi-steady stage as:

ust wRa2=5 x1=5 ðt=PÞ2=5 :

(41)

It is clear from Eqs. (40) and (41) that, while the surface is subject to ramp heating during the quasi-steady stage, the thickness of the thermal boundary layer shrinks with time whereas the velocity increases with time to maintain the balance between conduction and convection. This scenario is similar to the thermal boundary layer adjacent to a vertical wall subject to ramped heating, as described by Patterson et al. [34]. The growth of the velocity with time changes from t5/2 at the initial stage as specified in Eq. (27) to t2/5 at the quasi-steady stage described by scale (41). From scale (40) and (41), it is clear that as ramp finishes at time t ¼ P, the thickness of the thermal boundary layer reaches its minimum and the velocity reaches its maximum, and the values are the same as the respective steady state values of the constant heating case [15]. Therefore, the dividing position between the conductive and the convective subregions at the steady state follows the scaling in Mao et al. [15], which is (RaA5)1/3. As the ramp heating finishes at time P and the thermal forcing becomes constant afterwards, the quasi-steady stage soon transfers to a steady stage. Substituting t ¼ P into Eq. (41), the steady state velocity after P is obtained as

us wRa2=5 x1=5 :

(42)

This corresponds to a transition straight into the steady state given by Eq. (30). Comparing Eq. (42) with Eq. (30), and together with the findings of Section 4.2.1, it is revealed that for both the short and the long ramp scenarios, the scale of the steady state velocity for the ramped heating case is the same as that for the constant heating case despite that the flow undergoes different developments. Scale (40) predicts that during the quasi-steady state, the thickness of the thermal boundary layer is proportional to Ra1/5 and (t/P)1/5. To verify this, the temperature profiles along the vertical line of x ¼ 8.3 for different Rayleigh numbers and ramping durations are plotted in Fig. 10(a). Fig. 10(b) normalizes s by t/P, which is the ratio of the instantaneous temperature difference (between the water surface and the reference temperature) t/PDT at time t and the maximum temperature difference DT at time P. It is clear in Fig. 10(b) that, for the same Rayleigh number, the thickness of the thermal boundary layer decreases with time during the quasi-steady stage, and for the same time, the thickness decreases with the Rayleigh number, which agrees with the scaling prediction (40). Fig. 10(c) normalizes the depth with the thickness of the thermal boundary layer described by scale (40). It is clear in Fig. 10(c) that the different temperature profiles collapse together, confirming the correctness of scale (40).

Fig. 10. Verification of the thickness scale (40) for the thermal boundary layer. (a) Temperature profiles along the vertical line of x ¼ 8.3. (b) Temperature profiles with temperature normalized by t/P. (c) Temperature profiles with depth normalized by scale (40) and temperature normalized by t/P.

Apart from the thickness of the thermal boundary layer, the above analysis also reveals various flow developing stages (initial, quasi-steady, and steady) and the respective scales for different flow stages. Numerical simulations have been conducted to verify the existence of different stages and the respective scales for each stage. The time series of the maximum velocity along the vertical line of x ¼ 3.5 are plotted in Fig. 11 for the two possible flow scenarios of the ramp heating case along with the result of the constant heating case. Fig. 11 confirms that, irrespective of the ramp duration, the ramp heating leads to the same steady state velocity as the corresponding constant heating case. For the short ramp case of P < ts, as shown in Fig. 11(a), the flow velocity increases smoothly both before and after the ramp finishes and gradually reaches the steady state after time P. For the long ramp case of P > ts, as shown in Fig. 11(b), there is a distinct change in the growth curve of the flow velocity within the ramping duration. This change is due to the fact that a quasi-steady state is reached within the ramping duration. The significant variation of the velocity growth rate within the

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71

Fig. 11. The time series of horizontal velocity along the line of x ¼ 3.5 for the two different scenarios, the solid and dashed lines are the results of the ramp heating case and the constant heating case respectively. (a) The short ramp scenario P ¼ 0.04 < ts, Ra ¼ 106. (b) The long ramp scenario P ¼ 0.4 > ts, Ra ¼ 2  107.

ramping duration in Fig. 11(b) agrees well with the scaling prediction. As discussed above, the growth rate of the flow velocity decreases from t5/2 at the initial stage to t2/5 at the quasi-steady stage as revealed by scale (27) and (41) respectively, which are verified below. Since a balance between convection and conduction is maintained throughout the quasi-steady stage, the flow immediately becomes steady as soon as the ramp finishes at time P, as illustrated by Fig. 11(b). To further verify the scaling for different stages of the flow development for the long ramp scenario, results from the numerical simulation are normalized and plotted against the scaling prediction in Fig. 12. Scale (27) reveals that the velocity increases with time at a rate proportional to t5/2 during the initial stage of the thermal boundary layer growth, whereas scale (41) indicates that during the quasi-steady stage the velocity grows at a rate proportional to t2/5, i.e. at a much slower rate compared to the initial stage. To verify this, the time series of the maximum horizontal velocity umax along the local depth is shown in Fig. 12(a), and these time series are normalized with the proper scales in Eqs. (27) and (41) and plotted against t5/2 and t2/5 in Fig. 12(b) and (c) respectively. For the verification of the initial stage scaling (27), the time series of velocity is normalized with time-independent terms in Eq. (27) and plotted against t5/2 in Fig. 12(b). The collapse of different time series into a straight line at the very initial stage in Fig. 12(b) verifies both the time dependency and the Ra dependency of the velocity predicted by Eq. (27). Similarly, for the verification of the quasi-steady stage scaling in Eq. (41), the time series of velocity is normalized with the time-independent terms in Eq. (41) and plotted against t2/ 5 in Fig. 12(c). It is clear that for the velocity in the quasi-steady stage (immediately before the steady stage), different time series collapse onto a single straight line, confirming that the quasi-steady state flow velocity is linearly proportional to t2/5, as described in Eq. (41). To verify the steady state scaling of Eq. (42), the time series in Fig. 12(a) is normalized by scaling (42) and shown in Fig. 12(d). The

Fig. 12. Time series of umax along the vertical line of x ¼ 3.17. (a) Time series of umax. (b) Normalized umax from simulation plotted against t5/2. (c) Normalized umax from simulation plotted against t2/5. (d) Time series of umax normalized by the steady state velocity (42).

collapse of different time series at steady state clearly validates the steady state scaling. 5. Conclusions The present study has considered the unsteady natural convection in a wedge-shaped domain subject to ramped isothermal

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heating in which the surface temperature increases linearly with time over a certain time duration and then remains constant. Two distinct flow subregions are identified, namely, a conductive region and a convective region. The conductive region is investigated through a hybrid of analytical solution and scaling analysis, and the convective region is investigated through conventional scaling analysis. The analytical and scaling results are verified by numerical simulations. Different flow scenarios are revealed by comparing the ramp duration with the characteristic time scales of the flow development. For the conductive region, the characteristic time scale is the time for the thermal boundary layer to reach the local depth. For the convective region, it is the quasi-steady state time when convection balances conduction. (i) For the conductive region nearshore, a comparison between the ramp duration P and the time tsp for the thermal boundary layer to reach the local depth reveals two scenarios. For the short ramp scenario of P < tsp, the ramp finishes before the thermal boundary layer reaches the local depth. In this case, no steady or quasi-steady state is reached within the ramp duration, and after the ramp finishes, the flow velocity continues to increase until the thermal boundary layer reaches the local depth. The scale of the maximum flow velocity over the flow development is derived and verified. For the long ramp scenario of P > tsp, the ramp finishes after the thermal boundary layer reaches the local depth at time tsp. In this case, a quasi-steady state is present within the ramp duration after time tsp, i.e. over the period of tsp < t < P, during which the local temperature increases at the same rate as the water surface temperature and the flow velocity is steady. After the ramp finishes, i.e. t > P, the flow velocity and the temperature difference decrease with time. For both scenarios, the flow eventually becomes isothermal and stationary in the conductive region, the same as the constant heating case reported in Ref. [15]. (ii) For the convective region offshore, a comparison of the ramp duration P with the quasi-steady state time ts at which convection balances conduction also reveals two possible scenarios. For the short ramp scenario of P < ts, no quasi-steady state is reached within the ramp duration. For the long ramp scenario of P > ts, a quasi-steady state characterized by the balance of conduction and convection is reached at time ts within the ramp duration, and the flow becomes steady soon after the ramp finishes. The present scaling analysis reveals that, for the long ramp scenario, the dependency of the flow velocity on time changes from t5/2 at the initial stage to t2/5 at the quasi-steady stage, which is verified by the simulation results. For both scenarios, a steady velocity and a distinct thermal boundary layer are maintained in the final steady state, and the velocity at the final steady state for the ramp heating case is the same as that for the corresponding constant heating case. Therefore, the same destination is reached although the transient flow developments are different, and the transient flow development is determined by the comparison between the ramping duration P and the quasi-steady state time ts. With the above scales, some estimation of the order of magnitude in field situations can be derived. Assuming a temperature difference of DT ¼ 5  C, a water depth of h ¼ 1 m in field situations, a Rayleigh number around 7  1010 is expected based on the molecular properties of water. As derived in Mao et al. [15], the extent of the conductive region is confined to the nearshore region of x < (RaA5)1/3, corresponding to a dimensional value of 0.01 m

(assuming A ¼ 0.1), which is very close to shore. Therefore, most of the region falls into the convective subregion. For an idealized diurnal variation, the typical ramping duration P is of the order of 12 h. It can be derived from scale (34) that the time for the start of the quasi-steady state is of the order of 1 h at the maximum position of x ¼ 1/A. Therefore, the field situation falls into the long ramp scenario (ts < P) where a quasi-steady state is reached within the ramp duration after ts. In this case, the flow velocity grows with time at a rate proportional to t2/5 at the quasi-steady stage as described in scale (41) and verified in Fig. 12(c), reaching the maximum at the end of the ramp. The maximum flow velocity specified by Eq. (42) is the same as that for the corresponding constant heating case with the same Rayleigh number. Assuming a slope of A ¼ 0.1 and a horizontal extent of x ¼ 10 m, this corresponds to a maximum velocity of the order of 0.5 cm/s. It can be also demonstrated that the seasonal variation of ramp heating, which is over a much longer period, also falls into the long ramp scenario in which a quasi-steady state is reached within the ramp duration. As shown in Mao et al. [15], adopting the molecular values of viscosity and thermal diffusivity means that the thickness of the thermal boundary layer is confined to a thin surface layer of a few centimeters, which agrees with the field observations of James et al. [4] in the case of small wind-generated mixing. In the case of strong wind, vertical mixing is enhanced, leading to a thickened surface layer, which is also observed in the field experiment of James et al. [4]. In this case, eddy viscosity and diffusivity are more appropriate for the scaling, which will result in a significantly larger thickness. Apart from the vertical mixing, wind speed also strongly influences the magnitudes of sensible heat transfer and latent heat flux. As mentioned above, succession of the ramped heating with ramped cooling constitutes a cycle of a periodic variation, which is a step closer to the periodic thermal forcing in field situations. The present study considers the case when the surface temperature is warmer than the interior. If the temperature of the surface is lower than the interior, a surface instability may arise. Furthermore, for a periodic thermal forcing, a time lag between the flow response and the switch of the thermal forcing is expected, as already revealed in the field experiments [1,2]. The results of the present study suggest that the length of the periodic thermal forcing may determine the transient flow development and in turn the growth rate of the convective flow at different stages of the flow development.

Acknowledgement The authors are grateful for the financial support of the Australian Research Council. Yadan Mao also gratefully acknowledges the support of National Science Foundation of China (Grant Number 11002127) and the Fundamental Research Fund for National University, China University of Geosciences (Wuhan).

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