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Studies on Mixed Convection and Its Transition to Turbulence—A Review

  • Somenath Gorai
  • Sarit K. DasEmail author
Chapter
  • 150 Downloads

Abstract

Studies on mixed convective fluid flow and heat transfer are much more scarce compared to the large volume of literature available on either forced or natural convection. This is primarily because it was thought that applications of comparable forced and natural convection simultaneously are rather limited. However, the recent advent of high heat flux computing and LASER equipment and the need for their cooling has made mixed convection more relevant. The present review traces the development of studies in mixed convection over the last half a century. The most tricky and complex question in this respect may be that of the onset of turbulent flow in mixed convection. A clear and acceptable criterion for the transition of laminar flow to turbulent in this regime is still evasive. Hence, the review has culminated into a relook into the studies dedicated to these transition characteristics.

Keywords

Laminar mixed convection Turbulent mixed convection Transition of mixed convection 

Nomenclature

Dimensionless Numbers

\(Re\)

Reynolds number (\(\rho VD/\mu\))

\(Ri\)

Richardson number (\(Gr/Re^{2}\))

\(Pr\)

Prandtl number (\(\mu C_{p} /k\))

\(Ra\)

Rayleigh number (\(Gr.Pr\))

\(Nu\)

Nusselt number (\(hD/k\))

\(Kn\)

Knudsen number (\(\lambda /L\)); ‘\(\lambda\)’ is mean free path

\(Ha\)

Hartman number

\(\overline{Nu}\)

Average Nusselt number

\(Nu_{T}\)

Nusselt number for forced turbulent convection

\(Gr\)

Grashof number

\(Gr_{q}\)

Grashof number based on heat flux (\(g\beta D^{4} \dot{q}/\nu^{2} k\))

\(Gr_{D}\)

Grashof number based on diameter (\(g\beta D^{3} {\Delta }T/\nu^{2}\))

\(Gr_{L}\)

Grashof number based on constant wall temperature (\(g\beta L^{3} {\Delta }T/\nu^{2}\))

\(Gr_{T}\)

Thermal Grashof number (\(g\beta D^{3} {\Delta }T/\nu^{2}\))

\(Gr_{M}\)

Solutal Grashof number (\(g\beta_{M} D^{3} {\Delta \omega }/\nu^{2}\))

\(\overline{Ra}\)

Average Rayleigh number

\(\overline{Re}\)

Average Reynolds number

\(\widehat{Nu}\)

Weighted average Nusselt number

\(Gz\)

Graetz number \(\left( {\frac{D}{L}Re.Pr} \right)\)

\(Re_{cr}\)

Critical Reynolds number

\(Re_{\theta }\)

Transition momentum thickness Reynolds number

\(Re_{qt}\)

Reynolds value at the start of quasi-turbulent flow regime

\(Re_{x}\)

Local Reynolds number

\(Bo\)

Buoyancy parameter (\(Gr_{q} /Re^{m} Pr^{n}\))

\(Bo_{2}\)

Buoyancy parameter (\(Gr_{q} /Re^{2.5} Pr\))

\(K\)

Buoyancy parameter (\(Gr/Re^{2.5}\))

\(ER\)

Expansion ratio (outlet to inlet height ratio)

E

Enhancement ratio (ratio of heat input with porosity and without porosity)

Abbreviations

ACFD

Asymptotic Computational Fluid Dynamics

AWF

Analytical Wall Functions

CFD

Computational Fluid Dynamics

CMM

Compound Matrix Method

CMSIP

Coupled Modified Strongly Implicit Procedure

DNS

Direct Numerical Simulation

FCD

Forced Convection Developing

FD

Fully Developed

LBM

Lattice Boltzmann Method

LES

Large Eddy Simulation

MCD

Mixed Convection Developing

OpenFOAM

Open source Field Operation And Manipulation

PIV

Particle Image Velocimetry

RANS

Reynolds Averaged Navier-Stokes

RNG

Renormalization Group Method

SST

Shear Stress Transport

UHF

Uniform Heat Flux

UWT

Uniform Wall Temperature

Greek Letters

\(\varepsilon\)

Rate of dissipation of turbulence energy

\(\omega\)

Specific rate of dissipation

\(\varphi_{v}\)

Viscosity ratio (\(\mu_{b} /\mu_{w}\))

\(\mu_{b}\)

Dynamic viscosity at bulk temperature (Pa s)

\(\mu_{w}\)

Dynamic viscosity at the wall temperature (Pa s)

\(\lambda\)

Buoyancy parameter (\(Gr/Re^{2}\))

\(\emptyset\)

Volume fraction

\(\varphi\)

Angle (degree or radian)

\(\theta\)

Dimensionless temperature

\(\rho\)

Density (kg/m3)

\(\gamma\)

Intermittency in \(\gamma - Re_{\theta }\) transition model

\(\beta\)

Thermal expansion coefficient (K−1)

Other Symbols

k

Thermal conductivity (W/m K)

\(k\)

Turbulence kinetic energy (J)

Sp. Gr.

Specific gravity

g

Acceleration due to gravity (m/s2)

a, b, c

Constants in Eqs. 8, 12

\(L\)

Length (m)

\(D\)

Diameter (m)

\(D_{h}\)

Hydraulic diameter (m)

\(d_{e}\)

Equivalent diameter of a channel (m) (\(2hb/\left( {h + b} \right);\) \(h,b\) are channel height and width)

\(A\)

Aspect ratio (–)

Q

Heat rate (W)

\(\dot{q}\)

Heat flux (W/m2)

\(q_{1}\)

Cold wall heat flux (W/m2)

\(q_{2}\)

Hot wall heat flux (W/m2)

\(x, y\)

Axial and transverse coordinates (m)

\(u, v\)

Axial and transverse velocities (m/s)

\(X, Y\)

Dimensionless axial and transverse coordinates

\(U, V\)

Dimensionless axial and transverse velocity

\(x\)

Distance from the initial point of heating (m)

\(C_{p}\)

Specific heat at constant pressure (J/kg K)

\(C_{f}\)

Skin friction coefficient (–)

̴

Approximately

©

Copyright

Subscripts

lam, l

Laminar

b

Properties of fluid at bulk temperature

q

Based on heat flux

M

Mixed convection in Eq. 7

trans

Transition

\(c, cr\)

Critical value

\(lc\)

Laminar flow at critical Reynolds number

\(cr_{2}\)

Critical value at second wall

\(x\)

Local value

Superscripts

t

Turbulent

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Technology RoparRupnagar, PunjabIndia

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