Abstract
Background
In recent years, the average abundance function has attracted much attention as it reflects the degree of cooperation in the population. Then it is significant to analyse how ...average abundance functions can be increased to promote the proliferation of cooperative behaviour. However, further theoretical analysis for average abundance function with mutation under redistribution mechanism is still lacking. Furthermore, the theoretical basis for the corresponding numerical simulation is not sufficiently understood.
Results
We have deduced the approximate expressions of average abundance function with mutation under redistribution mechanism on the basis of different levels of selection intensity
$$\omega$$
ω
(sufficiently small and large enough). In addition, we have analysed the influence of the size of group
d
, multiplication factor
r
, cost
c
, aspiration level
$$\alpha$$
α
on average abundance function from both quantitative and qualitative aspects.
Conclusions
(1) The approximate expression will become the linear equation related to selection intensity when
$$\omega$$
ω
is sufficiently small. (2) On one hand, approximation expression when
$$\omega$$
ω
is large enough is not available when
r
is small and
m
is large. On the other hand, this approximation expression will become more reliable when
$$\omega$$
ω
is larger. (3) On the basis of the expected payoff function
$$\pi \left( \centerdot \right)$$
π
⋅
and function
$$h(i,\omega )$$
h
(
i
,
ω
)
, the corresponding results for the effects of parameters (
d
,
r
,
c
,
$$\alpha$$
α
) on average abundance function
$$X_{A}(\omega )$$
X
A
(
ω
)
have been explained.
We present an experimental and numerical study of turbulent thermal convection in the presence of an effective horizontal buoyancy that generates extra shear at the boundary. Geometrical confinements ...are also applied by varying the streamwise and spanwise aspect ratios of the convection cell to condense the plumes. With these, we systematically explore the effects of plume and shear on heat transfer. It is found that a streamwise confinement results in increased plume coverage but decreased shear compared with spanwise confinement. The fact that streamwise confinement leads to a higher vertical heat transfer efficiency than the spanwise confined case suggests that the increase of plume coverage is the dominant effect responsible for the enhanced heat transfer. Our results highlight the potential applications of coherent structure manipulation in efficient passive heat transfer control and thermal engineering. We also analyse the energetics of the present system and derive the expression of mixing efficiency accordingly. The mixing efficiency is found to increase with both the buoyancy ratio and streamwise dimension.
A primary objective in turbulent thermal convection research is to understand and control the heat transport scaling behaviour. Previous studies have shown that the heat transport can be tuned by ...manipulating the boundary layer topographies with monoscale roughness elements. Now, Zhu et al. (J. Fluid Mech., vol. 869, 2019, R4) have demonstrated that with multiscale wall roughness, the heat transport law with an exponent of
$1/2$
can be achieved for an extended range of the Rayleigh number, providing a new way to manipulate heat transport by tuning boundary topographies in turbulent flows.
We report an experimental study of the viscous boundary layer (BL) properties of turbulent Rayleigh–Bénard convection in a cylindrical cell. The velocity profile with all three components was ...measured from the centre of the bottom plate by an integrated home-made particle image velocimetry system. The Rayleigh number $Ra$ varied in the range $1.82 \times 10^8 \le Ra \le 5.26 \times 10^9$ and the Prandtl number $Pr$ was fixed at $Pr = 4.34$. The probability density function of the wall-shear stress indicates that using the velocity component in the mean large-scale circulation (LSC) plane alone may not be sufficient to characterise the viscous BL. Based on a dynamic wall-shear frame, we propose a method to reconstruct the measured full velocity profile which eliminates the effects of complex dynamics of the LSC. Various BL properties including the eddy viscosity are then obtained and analysed. It is found that, in the dynamic wall-shear frame, the eddy viscosity profiles along the centre line of the convection cell at different $Ra$ all collapse on a single master curve described by $\nu _t^d / \nu = 0.81 (z / \delta _u^d) ^{3.10 \pm 0.05}$. The Rayleigh number dependencies of several BL quantities are also determined in the dynamic frame, including the BL thickness $\delta _u^d$ (${\sim } Ra^{-0.21}$), the Reynolds number $Re^d$ (${\sim }Ra^{-0.46}$) and the shear Reynolds number $Re_s^d$ (${\sim } Ra^{0.24}$). Within the experimental uncertainty, these scaling exponents are the same as those obtained in the static laboratory frame. Finally, with the measured full velocity profile, we obtain the energy dissipation rate at the centre of the bottom plate $\varepsilon _{w}$, which is found to follow $\langle \varepsilon _{w} \rangle _t \sim Ra^{1.25}$.
We present a systematic investigation of the effects of roughness geometry on turbulent Rayleigh–Bénard convection (RBC) over rough plates with pyramid-shaped and periodically distributed roughness ...elements. Using a parameter
$\unicodeSTIX{x1D706}$
defined as the height of a roughness element over its base width, the heat transport, the flow dynamics and the local temperatures are measured for the Rayleigh number range
$7.50\times 10^{7}\leqslant Ra\leqslant 1.31\times 10^{11}$
and Prandtl numbers
$Pr$
from 3.57 to 23.34 at four values of
$\unicodeSTIX{x1D706}$
(0.5, 1.0, 1.9 and 4.0). It is found that the heat transport scaling, i.e.
$Nu\sim Ra^{\unicodeSTIX{x1D6FC}}$
where
$Nu$
is the Nusselt number, may be classified into three regimes in turbulent RBC over rough plates. In Regime I, the system is in a dynamically smooth state. The heat transport scaling is the same as that in a smooth cell. In Regimes II and III, the heat transport is enhanced. When
$\unicodeSTIX{x1D706}$
is increased from 0.5 to 4.0,
$\unicodeSTIX{x1D6FC}$
increases from 0.36 to 0.59 in Regime II and it increases from 0.30 to 0.50 in Regime III. The experiment thus clearly demonstrates that the heat transport scaling in turbulent RBC can be manipulated using
$\unicodeSTIX{x1D706}$
in the heat transport enhanced regime. Previous studies suggest that the transition to heat transport enhanced regime, i.e. from Regime I to Regime II, occurs when the thermal boundary layer (BL) thickness becomes smaller than the roughness height. Direct measurements of the viscous BL in the present study suggest that the transition from Regime II to Regime III is likely a result of the viscous BL thickness becoming smaller than the roughness height. The scaling exponent of the Reynolds number
$Re$
with respect to
$Ra$
changes from 0.471 to 0.551 when
$\unicodeSTIX{x1D706}$
is increased from 0.5 to 4.0, suggesting a change of the dynamics of the large-scale circulation. Interestingly, the transition from Regime II to Regime III in terms of the heat transport scaling is not reflected in the
$Re$
scaling with
$Ra$
. It is also found that increasing
$\unicodeSTIX{x1D706}$
increases the clustering of thermal plumes which effectively increases the plume lifetime. This leads to a great increase in the probability of observing large temperature fluctuations in the bulk flow, which corresponds to the formation of more coherent plumes or plume clusters that are ultimately responsible for the enhanced heat transport.
We consider the situation of a misalignment between the global temperature gradient and gravity in thermal convection. In such a case an effective horizontal buoyancy arises that will significantly ...influence the transport properties of heat, mass and momentum. It may also change the flow morphology in turbulent convection. In this paper, we present an experimental and numerical study, using Rayleigh–Bénard convection as a platform, to explore systematically the effect of horizontal buoyancy on heat transport in turbulent thermal convection. Experimentally, a condition of increasing horizontal Rayleigh number ($Ra_H$, which is the non-dimensional horizontal thermal driving strength) under fixed vertical Rayleigh number ($Ra_V$, the non-dimensional vertical driving strength) is achieved by tilting the convection cell and simultaneously increasing the imposed temperature difference. We find that, with increasing horizontal to vertical buoyancy ratio ($\varLambda = Ra_H/Ra_V$), the overall heat transport manifests a monotonic increase in vertical heat transport ($Nu_V$) as well as a monotonic increase in its horizontal component ($Nu_H$). However, the horizontal Nusselt number is found to be approximately one order of magnitude smaller than the vertical Nusselt for the parameter range explored. We also show that the non-zero $Nu_H$ results from the broken azimuthal symmetry of the system induced by the horizontal buoyancy. We find that the enhancement of vertical heat transport comes from the increased shear generated by the horizontal buoyancy at the boundary layer. The effect of Prandtl number ($Pr$) is also studied numerically. Finally, we extend the Grossmann–Lohse theory to the case with an effective horizontal buoyancy, the result of which is successful in predicting $Nu_V(Ra_V,\varLambda ,Pr)$.
A novel experiment was performed in rotating Rayleigh–Bénard convection (RRBC), wherein the convection cell with radius $R$ was shifted away from the rotation axis by a distance $d$. In this case, ...the centrifugal force felt by a fluid parcel (characterized by the Froude number $Fr$) can be decomposed into an axisymmetrical part $Fr_R$ and a directed one $Fr_d$. It has been reported that the global heat transport enhances at $Fr_{d,c}$ and then reaches an optimal state at $Fr_{d,max}$ (Hu et al., Phys. Rev. Lett., vol. 127, 2021, 244501). In this paper, the local properties after the offset effects set in are investigated further, which show different features before and after $Fr_{d,max}$. The local temperature measurements at the cell centre reveal that the bulk flow turns from a turbulent state into a laminar state at $Fr_{d,max}$, which is consistent with the particle image velocimetry results. This transition can be qualitatively understood by an equivalent tilted RRBC system. As for the hot and cold coherent structures near the sidewall, their vertical temperature variations reach a minimum at $Fr_{d,max}$, implying that these structures are mostly uniform in the vertical direction at $Fr_{d,max}$. Their temperature contrasts show a linear dependence on $Fr_d$ and start to deviate from this linear behaviour when $Fr_d>Fr_{d,max}$. Besides the dominant effects of $Fr_d$, the secondary effects of $Fr_R$ are also investigated. Due to the positive effect of $+Fr_R$ on the cold structure and the negative effect of $-Fr_R$ on the hot one, the cold structure is more coherent than the hot one, but its size is smaller. The shift of the cold cluster centre from the farthest point is also larger than the shift of the hot one from the nearest point.
We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio
$\unicodeSTIX{x1D6E4}$
,
$1/128\leqslant ...\unicodeSTIX{x1D6E4}\leqslant 1$
, and Rayleigh number
$Ra$
,
$3\times 10^{4}\leqslant Ra\leqslant 1\times 10^{11}$
, at a fixed Prandtl number of
$Pr=4.38$
by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing
$\unicodeSTIX{x1D6E4}$
from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number
$Nu$
, exhibits the scaling
$Nu-1\sim Ra^{0.61}$
over three decades of
$Ra$
at
$\unicodeSTIX{x1D6E4}=1/128$
, which contrasts sharply with the classical scaling
$Nu-1\sim Ra^{0.31}$
found at
$\unicodeSTIX{x1D6E4}=1$
. The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition,
$Ra^{\ast }=(29.37/\unicodeSTIX{x1D6E4})^{3.23}$
, such that the scaling transition in
$Nu$
and
$Re$
can be clearly demonstrated when plotted against
$Ra/Ra^{\ast }$
.
The Rayleigh number (
$\mathit{Ra}$
) scaling of the global Bolgiano length scale
${L}_{B, global} $
and the local Bolgiano length scale
${L}_{B, centre} $
in the centre region of turbulent ...Rayleigh–Bénard convection are investigated for Prandtl numbers
$\mathit{Pr}= 0. 7$
and
$4. 38$
and
$3\times 1{0}^{5} \leq \mathit{Ra}\leq 3\times 1{0}^{9} $
. It is found that
${L}_{B, centre} $
does not necessarily exhibit the same scaling as
${L}_{B, global} $
. While
${L}_{B, global} $
is monotonically deceasing as
${L}_{B, global} \sim {\mathit{Ra}}^{- 0. 10} $
for both
$\mathit{Pr}$
,
${L}_{B, centre} $
shows a steep increase beyond a certain
$\mathit{Ra}$
value. The complex scaling of the local Bolgiano length scale in the centre is a result of the different behaviour of the temperature-variance dissipation rate,
${\epsilon }_{T} $
, and the turbulent-kinetic-energy dissipation rate,
${\epsilon }_{u} $
. This shows that for sufficiently high
$\mathit{Ra}$
the flow is well-mixed and hence temperature is passively advected. It is also observed that the
$\mathit{Ra}$
-range in which
${L}_{B, centre} $
exhibits the same scaling as the global Bolgiano length scale is increasing with increasing
$\mathit{Pr}$
. It is further observed that for
$\mathit{Pr}= 4. 38$
and
$\mathit{Ra}\leq 3\times 1{0}^{7} $
the local vertical heat flux in the centre region is balanced by the turbulent-kinetic-energy dissipation rate. For higher
$\mathit{Ra}$
we find that the local heat flux is decreasing. At
$\mathit{Pr}= 0. 7$
we do not observe such a balance, as the measured heat flux is between the heat fluxes estimated through the turbulent-kinetic-energy dissipation rate and the temperature-variance dissipation rate. We therefore suggest that the balance of the local heat flux might be Prandtl-number dependent. The conditional average of the local vertical heat flux
$\mathop{\langle \mathit{Nu}\vert {\epsilon }_{u} , {\epsilon }_{T} \rangle }\nolimits_{\mathit{centre}} $
in the core region of the flow reveals that the highest vertical heat flux occurs for rare events with very high dissipation rates, while the joint most probable dissipation rates are associated with very low values of vertical heat flux. It is also observed that high values of
${\epsilon }_{u} $
and
${\epsilon }_{T} $
tend to occur together. It is further observed that the longitudinal velocity structure functions approach Kolmogorov K41 scaling. The temperature structure functions appear to approach Bolgiano–Obukhov BO59 scaling for
$r\gt {L}_{B, centre} $
, while a scaling exponent smaller than the BO59 scaling is observed for separations
$r\lt {L}_{B, centre} $
. The mixed velocity and temperature structure function for
$\mathit{Ra}= 1\times 1{0}^{9} $
and
$\mathit{Pr}= 4. 38$
shows a short
$4/ 5$
-scaling for
$r\gt {L}_{B, centre} $
. Our results suggest that BO59 scaling might be more clearly observable at higher Prandtl and moderate Rayleigh numbers.
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