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Investigation of obstacle effect to improve conjugate heat transfer in backward facing step channel using fast simulation of incompressible flow

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Abstract

The current study investigates the conjugate heat transfer characteristics for laminar flow in backward facing step channel. All of the channel walls are insulated except the lower thick wall under a constant temperature. The upper wall includes a insulated obstacle perpendicular to flow direction. The effect of obstacle height and location on the fluid flow and heat transfer are numerically explored for the Reynolds number in the range of 10 ≤ Re  ≤ 300. Incompressible Navier-Stokes and thermal energy equations are solved simultaneously in fluid region by the upwind compact finite difference scheme based on flux-difference splitting in conjunction with artificial compressibility method. In the thick wall, the energy equation is obtained by Laplace equation. A multi-block approach is used to perform parallel computing to reduce the CPU time. Each block is modeled separately by sharing boundary conditions with neighbors. The developed program for modeling was written in FORTRAN language with OpenMP API. The obtained results showed that using of the multi-block parallel computing method is a simple robust scheme with high performance and high-order accurate. Moreover, the obtained results demonstrated that the increment of Reynolds number and obstacle height as well as decrement of horizontal distance between the obstacle and the step improve the heat transfer.

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Abbreviations

A, B :

Jacobian matrixes

c :

Pseudo-speed of sound

eff:

Efficiency of parallel computing

E :

Flux vector in x-direction

F :

Flux vector in y-direction

h :

Heat transfer coefficient, (W/m2.K)

H :

Dimensionless channel width

H w :

Dimensionless height of thick wall

H o :

Dimensionless height of obstacle

I :

Unit matrix

k :

Thermal conductivity, (W/m .K)

L w :

Dimensionless length of thick wall

N :

Number of blocks

Nu:

Nusselt number, \( h\widehat{H}/k \)

P :

Dimensionless pressure

Pr:

Prandtl number, ν/α

Q :

Vector of primitive variable

Re:

Reynolds number, \( \rho \widehat{H}\widehat{u}/\mu \)

SF :

Speed-up factor

t 1 :

Consumed time for a single block, (s)

t N :

Consumed time for N blocks, (s)

T :

Temperature, (K)

u :

Dimensionless velocity component in x-direction

v :

Dimensionless velocity component in y-direction

X o :

Dimensionless location of obstacle

X :

x- axis

Y :

y- axis

α :

Thermal diffusivity, (m2/s)

β :

Artificial compressibility factor

Δ :

Difference

μ :

Dynamic viscosity (N.s/m2)

θ :

Dimensionless temperature, (T − T i )/(T w  − T i )

ρ :

Density, (kg/m3)

τ :

Pseudo-time

ν :

Kinematic viscosity (m2/s)

ave:

Average

diff:

Diffusion

f:

Fluid

in :

Inlet

max :

Maximum

o :

Obstacle

s:

Solid

v :

Viscous

w :

Wall

^:

Dimensional variables in Eq. (4)

References

  1. Patankar SV, Spalding DB (1972) A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 15(10):1787–1806

    Article  MATH  Google Scholar 

  2. Chorin AJ (1967) A numerical method for solving incompressible viscous flow problems. J Comput Phys 2(1):12–26

    Article  MATH  Google Scholar 

  3. Rogers SE, Kwak D (1990) Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA J 28(2):253–262

    Article  MATH  Google Scholar 

  4. Rogers SE, Kwak D (1991) An upwind differencing scheme for the incompressible Navier–strokes equations. Appl Numer Math 8(1):43–64

    Article  MathSciNet  MATH  Google Scholar 

  5. Shah A, Yuan L (2009) Flux-difference splitting-based upwind compact schemes for the incompressible Navier–Stokes equations. Int J Numer Methods Fluids 61(5):552–568

    Article  MathSciNet  MATH  Google Scholar 

  6. Shah A, Guo H, Yuan L (2009) A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier–Stokes equations. Commun Comput Phys 5(2–4):712–729

    MathSciNet  MATH  Google Scholar 

  7. Meyers J, Lacor C, Baelmans M (2008) On the use of high-order finite-difference discretization for LES with double decomposition of the subgrid-scale stresses. Int J Numer Methods Fluids 56(4):383–400

    Article  MathSciNet  MATH  Google Scholar 

  8. Christie I (1985) Upwind compact finite difference schemes. J Comput Phys 59(3):353–368

    Article  MATH  Google Scholar 

  9. Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhong X (1998) High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition. J Comput Phys 144(2):662–709

    Article  MathSciNet  MATH  Google Scholar 

  11. Nouri-Borujerdi A, Kebriaee A (2012) Upwind compact implicit and explicit high-order finite difference schemes for level set technique. Int J Comput Methods Eng Sci Mech 13(4):308–318

    Article  Google Scholar 

  12. Kondoh T, Nagano Y, Tsuji T (1993) Computational study of laminar heat transfer downstream of a backward-facing step. Int J Heat Mass Transf 36(3):577–591

    Article  MATH  Google Scholar 

  13. Kumar A, Dhiman AK (2012) Effect of a circular cylinder on separated forced convection at a backward-facing step. Int J Therm Sci 52:176–185

    Article  Google Scholar 

  14. Selimefendigil F, Öztop HF (2013) Numerical analysis of laminar pulsating flow at a backward facing step with an upper wall mounted adiabatic thin fin. Comput Fluids 88:93–107

    Article  Google Scholar 

  15. Kanna PR, Das MK (2006) Conjugate heat transfer study of backward-facing step flow–a benchmark problem. Int J Heat Mass Transf 49(21):3929–3941

    Article  MATH  Google Scholar 

  16. Teruel FE, Fogliatto E (2013) Mecánica Computacional, Volume XXXII. Number 39. Heat and Mass Transfer (C)

  17. Ramšak M (2015) Conjugate heat transfer of backward-facing step flow: a benchmark problem revisited. Int J Heat Mass Transf 84:791–799

    Article  Google Scholar 

  18. Bhumkar YG, Sheu TW, Sengupta TK (2014) A dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations. J Comput Phys 278:378–399

    Article  MathSciNet  MATH  Google Scholar 

  19. Weatherill N, Forsey C (1984) Grid generation and flow calculations for complex aircraft geometries using a multi-block scheme. Rep/AIAA

  20. Durst F, Schäfer M (1996) A parallel block-structured Multigrid method for the prediction of incompressible flows. Int J Numer Methods Fluids 22(6):549–565

    Article  MATH  Google Scholar 

  21. Drikakis D (1996) A parallel multiblock characteristic-based method for three-dimensional incompressible flows. Adv Eng Softw 26(2):111–119

    Article  Google Scholar 

  22. Drikakis D, Iliev O, Vassileva D (1998) A nonlinear multigrid method for the three-dimensional incompressible Navier–Stokes equations. J Comput Phys 146(1):301–321

    Article  MATH  Google Scholar 

  23. Parikh P (2001) Application of a scalable, parallel, unstructured-grid-based Navier-Stokes solver. AIAA paper 2584

  24. Jia R, Sundén B (2003) Multiblock implementation strategy for a 3-D pressure-based flow and heat transfer solver. Numerical Heat Transfer: Part B: Fundamentals 44(5):457–472

    Article  Google Scholar 

  25. Chao J, Haselbacher A, Balachandar S (2009) A massively parallel multi-block hybrid compact–WENO scheme for compressible flows. J Comput Phys 228(19):7473–7491

    Article  MathSciNet  MATH  Google Scholar 

  26. Fico V, Emerson DR, Reese JM (2011) A parallel compact-TVD method for compressible fluid dynamics employing shared and distributed-memory paradigms. Comput Fluids 45(1):172–176

    Article  MATH  Google Scholar 

  27. Ghadimi M, Farshchi M (2012) Fourth order compact finite volume scheme on nonuniform grids with multi-blocking. Comput Fluids 56:1–16

    Article  MathSciNet  MATH  Google Scholar 

  28. Hoffmann KA, Chiang ST (2000) {Computational fluid dynamics, Vol. 1}. Wichita, KS: Engineering Education System

  29. Agarwal R (1981) A third-order-accurate upwind scheme for Navier-Stokes solutions at high Reynolds numbers. In: 19th Aerospace Sciences Meeting, p 112

  30. Fu D, Ma Y, Kobayashi T (1996) Nonphysical oscillations in numerical solutions: reason and improvement. CFD J 4(4):427–450

    Google Scholar 

  31. Fu D, Ma Y (1997) A high order accurate difference scheme for complex flow fields. J Comput Phys 134(1):1–15

    Article  MathSciNet  MATH  Google Scholar 

  32. Roe PL (1981) Approximate Riemann solvers, parameter vectors, and difference schemes. J Comput Phys 43(2):357–372

    Article  MathSciNet  MATH  Google Scholar 

  33. Hartwich P-M, Hsu C-H (1986) An implicit flux-difference splitting scheme for three-dimensional, incompressible Navier-Stokes solutions to leading-edge vortex flows. AIAA paper 86:1839

    Google Scholar 

  34. Beam RM, Warming R (1978) An implicit factored scheme for the compressible Navier-Stokes equations. AIAA J 16(4):393–402

    Article  MATH  Google Scholar 

  35. Malan A, Lewis R, Nithiarasu P (2002) An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: part I. Theory and implementation. Int J Numer Methods Eng 54(5):695–714

    Article  MATH  Google Scholar 

  36. Sheng C, Whitfield DL (1999) Multiblock approach for calculating incompressible fluid flows on unstructured grids. AIAA J 37(2):169–176

    Article  Google Scholar 

  37. Quinn MJ (2003) Parallel Programming, vol 526. TMH CSE

  38. McCool MD, Robison AD, Reinders J (2012) Structured parallel programming: patterns for efficient computation. Elsevier

  39. Gartling DK (1990) A test problem for outflow boundary conditions—flow over a backward-facing step. Int J Numer Methods Fluids 11(7):953–967

    Article  Google Scholar 

  40. Lê TH, Troff B, Sagaut P, Dang-Tran K, Phuoc LT (1997) PEGASE: a Navier-Stokes solver for direct numerical simulation of incompressible flows. Int J Numer Methods Fluids 24(9):833–861

    Article  MathSciNet  MATH  Google Scholar 

  41. Armaly BF, Durst F, Pereira J, Schönung B (1983) Experimental and theoretical investigation of backward-facing step flow. J Fluid Mech 127:473–496

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    Ali Nouri-Borujerdi

  2. Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran

    Arash Moazezi

Authors
  1. Ali Nouri-Borujerdi
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  2. Arash Moazezi
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Correspondence to Arash Moazezi.

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Nouri-Borujerdi, A., Moazezi, A. Investigation of obstacle effect to improve conjugate heat transfer in backward facing step channel using fast simulation of incompressible flow. Heat Mass Transfer 54, 135–150 (2018). https://doi.org/10.1007/s00231-017-2086-4

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  • DOI: https://doi.org/10.1007/s00231-017-2086-4

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