# Investigation of heat transfer and pressure drop of turbulent flow in tubes with successive alternating wall deformation under constant wall temperature boundary conditions

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## Abstract

In this study, the effects of the presence of multi-longitudinal vortex and various rotation angles between the pitches of alternating elliptical axis (AEA) tubes on heat transfer and pressure drop of turbulent flow were numerically investigated. Turbulent flow of water fluid was simulated at Reynolds numbers of 10,000–60,000. The turbulent flow and heat transfer in the tubes were discussed in terms of parameters such as static pressure, velocity magnitude, wall shear stress, turbulent intensity, performance evaluation criterion and the field synergy principle. The results demonstrate that most heat transfer occurs in the transition zone, but this also caused a high rate of pressure drop. Increasing the rotation angle between pitches from 60° to 80° increased the heat transfer, which increased the number of the multi-longitudinal vortices from four to eight and better mixed the cold fluid with the hot fluid near the tube wall on more paths. The friction factor decreased and the average Nusselt number increased as the Reynolds number increased. Both parameters increased as the angle of the pitch rotation increased. The performance evaluation criteria for all AEA tubes at a constant pumping power showed that the highest value (1.09) was achieved at a low Reynolds number (*Re* = 10,000) in the AEA 90° tube.

### Keywords

Turbulent flow Heat transfer Multi-longitudinal vortex Field synergy principle Performance evaluation criterion Alternating elliptical axis tube### Nomenclature

*A*Area, m

^{2}- A
Major axes length of elliptical cross section, mm

- B
Minor axes length of elliptical cross section, mm

- C
Transition length, mm

*C*_{p}Specific heat, J/(kg K)

*C*Perimeter of the ellipse, m

*D*_{h}Hydraulic diameter, m

*d*Circular tube diameter of AEA tube, mm

*f*Friction factor, \(f = {{(\Delta p D_{\text{h}} )} \mathord{\left/ {\vphantom {{(\Delta p D_{\text{h}} )} {\left( {\frac{1}{2}\rho u_{\text{avg}}^{2} L} \right)}}} \right. \kern-0pt} {\left( {\frac{1}{2}\rho u_{\text{avg}}^{2} L} \right)}}\)

*G*_{k}Production of turbulence kinetic energy, kg/(m s

^{3})*g*Gravitational acceleration, m/s

^{2}*h*_{loss}Irreversible head loss, m

*K*Thermal conductivity, W/(m K)

*k*Turbulence kinetic energy, m

^{2}/s^{2}*L*Length of tube, mm

*L*_{d}Length of inlet and outlet circular tube, mm

*Nu*Nusselt number, \(Nu = (q^{\prime\prime}D_{\text{h}} )/(K (T_{\text{w}} - T_{\text{b}} ))\)

*P*Pressure, kg/(m s

^{2})- P
Pitch length, mm

- \(\overline{P}\)
Mean pressure, kg/(m s

^{2})*P*_{k}Improved production of turbulence kinetic energy, kg/(m s

^{3})*Pr*Prandtl number,

*Pr*=*µC*_{p}/*K**q*″Heat flux, W/m

^{2}*Re*Reynolds number, Re =

*ρuD*_{ h }/*µ**S*Strain tensor magnitude, s

^{−1}*S*_{ij}Components of the mean strain tensor, s

^{−1}*T*Temperature, K

*T*_{b}Average bulk temperature, K

- \(\overline{T}\)
Mean temperature, K

*T*′Turbulent temperature fluctuations, K

*u*_{i}Velocity, m/s

- \(\overline{{u_{i} }}\)
Mean velocity, m/s

*u*_{i}′Turbulent velocity fluctuations, m/s

*u**Friction velocity, m/s

*V*Velocity, m/s

*x*_{i}Cartesian coordinates, m

*Y*Distance of the closest computational node from the wall, m

*y*^{+}Dimensionless wall distance

*z*Axial distance from inlet, m

### Greek symbols

*α*Kinetic energy correction factors

*δ*_{ij}Kronecker delta

*ɛ*Turbulent dissipation rate, m

^{2}/s^{3}*ɛ*_{imn}Tensor of Levi–Civita

*θ*Angle between major axes of elliptical cross-sectional tubes

*μ*Laminar dynamic viscosity, kg/(m s)

*μ*_{t}Turbulent dynamic viscosity, kg/(m s)

*μ*_{eff}Effective dynamic viscosity, kg/(m s)

*ν*Kinematic viscosity, m

^{2}/s*ρ*Density, kg/m

^{3}*σ*_{k}Turbulent Prandtl number of

*k**σ*_{ɛ}Turbulent Prandtl number of

*ɛ**τ*_{ij}Stress tensor, kg/(m s

^{2})*τ*_{w}Wall shear stress, kg/(m s

^{2})*Ω*Characteristic rotation rate, s

^{−1}*Ω*_{ij}Components of the vorticity tensor, s

^{−1}*Ω*_{m}^{rot}Components of the system rotation vector, s

^{−1}*ω*Turbulence eddy frequency, s

^{−1}

### Subscripts

*a*Section A

- avg
Average

*b*Section B

- eff
Effective

- s
Smooth tube

- w
Wall

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