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Reynolds analogy and turbulent heat transfer on rotating curved surfaces

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Abstract

To evaluate turbulent heat transfer on rotating turbine blades, knowledge of the Reynolds stress and heat conduction moments on the blades is required. This involves solving the mean flow equations of momentum and thermal energy on the rotating blades. In this paper, the turbine blades are represented by rotating curved surfaces, and turbulent heat transfer calculations are carried out by invoking a valid Reynolds analogy because of its simplicity. The classic Reynolds analogy is defined as the ratio between the eddy and thermal diffusivity in a stationary plane flow; however, its counterpart for rotating curved flows is still not known. To derive a Reynolds analogy for rotating curved flows, appropriate turbulence models to close the Reynolds transport equations for momentum and thermal energy are required. Assuming that, in the constant flux region, advection and diffusion of Reynolds stress and heat conduction moments are negligible compared to the production of turbulent energy, the Reynolds shear stresses and heat conduction moments can be solved in terms of the mean flow gradients, a turbulent Prandtl number (Prt)rc and a gradient Richardson number for rotating curved flows Rirc. These dimensionless numbers are dependent on mean flow properties, surface curvature, and system rotational speed. Thus derived, the classic Reynolds analogy can be recovered only when Rirc << 1 and (Prt)rc = 1. For turbine blade heat transfer, Rirc is not necessary very small and (Prt)rc differs from unity in the whole flow field. Therefore, using the classic Reynolds analogy to estimate turbine blade heat transfer will lead to incorrect result. The new Reynolds analogy, formulated for turbulent thermal flows on rotating curved surfaces, takes all pertinent external body force effects into account; thus, turbulent heat transfer evaluation on turbine blade surfaces can be substantially improved.

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Abbreviations

B:

integration constant: Eq. (63)

C:

integration constant: Eq. (64)

C f :

skin friction coefficient

C v :

specific heat of fluid at constant volume

C p :

specific heat of fluid at constant pressure

h =  1 +  ky :

metric coefficient

i, j, k, l, m etc.:

subscript indices used to designate vectors and tensors

k =  R−1(x):

surface curvature

LB :

Monin-Oboukhov length for stratified flow: Eq. (57)

LC :

Monin-Oboukhov length for stationary curved flows: Eq. (69)

LR :

Monin-Oboukhov length for rotating plane flows: Eq. (70)

Lrc :

Monin-Oboukhov length for rotating curved flow: Eq. (59)

\( \mathrm{P}=\kern0.5em {\mathrm{P}}^{\prime }-\kern0.5em \frac{1}{2}\ {\Omega}^2{r}^2 \) :

Reduced mean static pressure

P′ :

mean static pressure

p:

fluctuating static pressure

\( {\mathit{\Pr}}_t=\frac{\varepsilon_{\mathrm{M}}}{\varepsilon_{\mathrm{H}}} \) :

turbulent Prandtl number for stationary plane flow

(Prt)c :

turbulent Prandtl number for stationary curved flow

\( {\left({\mathit{\Pr}}_t\right)}_{rc}=\kern0.5em \frac{{\overset{\sim }{\epsilon}}_M}{{\overset{\sim }{\epsilon}}_H} \) :

turbulent Prandtl number for rotating curved flow

\( {\mathrm{q}}^2=\kern0.5em \overline{{\mathrm{u}}^2}+\kern0.5em \overline{{\mathrm{v}}^2}+\kern0.5em \overline{{\mathrm{w}}^2} \) :

kinetic energy of turbulence

\( {\dot{\mathrm{q}}}_w \) :

wall heat flux per unit area

r:

radial distance from origin of coordinate system

R(x):

radius of curvature of surface

Ri :

classic Richardson number

Ri c :

gradient Richardson number for stationary curved flows

Ri rc :

gradient Richardson number for rotating curved flows: Eq. (35)

St:

Stanton number: Eq. (77)

T:

mean temperature of flow

Tw :

mean temperature at the wall

\( {\mathrm{T}}^{+}=\kern0.5em \frac{C_p\left({\mathrm{T}}_w-\kern0.5em \mathrm{T}\right){\tau}_w}{{\mathrm{u}}_{\tau }\ {\dot{\mathrm{q}}}_w}\kern1em \) :

normalized mean temperature

U:

mean velocity along x-axis

Ui :

ith component of mean velocity

Upw :

mean velocity evaluated from wall static pressure measurements

ui :

ith component of fluctuating velocity

uτ =  (τw/ρ)1/2 :

friction velocity evaluated at the wall

u+ =  U/uτ :

normalized mean velocity

u,   v,   w:

fluctuating velocities along x, y, z directions, respectively

y+ =  y uτ/ν :

normalized y coordinate

α :

constant: Eq. (30)

β :

constant: Eq. (31)

\( {\beta}_1=\kern0.5em \frac{3}{8}\beta \) :

(constant)

γ =  ℓ1/ℓ2 :

length scale ratio

ε ijk :

the alternating tensor

ε M :

eddy diffusivity for stationary plane flows

\( {\overset{\sim }{\varepsilon}}_M \) :

eddy diffusivity for rotating curved flows

(εM)c :

eddy diffusivity for stationary curved flows

ε H :

thermal diffusivity for stationary plane flows

\( {\overset{\sim }{\varepsilon}}_H \) :

thermal diffusivity for rotating curved flows

(εH)c :

thermal diffusivity for stationary curved flows

θ:

fluctuating temperature

κ :

von Karman constant

κ f :

thermal conductivity of fluid

ℓ o :

mixing length for stationary plane flows

ℓ 1 :

length scale: Eq. (9)

ℓ 2 :

length scale: Eq. (11)

Λ:

dissipation length scale: Eq. (10)

μ :

fluid viscosity

ν :

fluid kinematic viscosity

ρ :

fluid density

σ =  κf/ρ Cp :

molecular thermal diffusivity

τ :

shear stress

τ w :

wall shear stress

Ω:

rotational speed of coordinate system about z-axis

Ωi :

ith component of rotation vector

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Acknowledgements

RMCS is deeply appreciative of the data provided him by various authors of the different cases used to compare with simulation results derived from the current Reynolds analogy. The author is also greatly indebted to Mr. Sky K. W. Tse for helping to create the figures shown in the paper.

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  1. Mechanical Engineering Department, The Hong Kong Polytechnic University, Hung Hom, Kowloon, HKSAR, Hong Kong

    R. M. C. So

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So, R.M.C. Reynolds analogy and turbulent heat transfer on rotating curved surfaces. Heat Mass Transfer 56, 2813–2830 (2020). https://doi.org/10.1007/s00231-020-02901-1

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