## Abstract

This paper presents the numerical modeling of complex flows and heat transfer. The Finite Analytic method is used to discretize the transport equations. The diagonal Cartesian method is proposed to model fluid flows and heat transfer over complex geometries. A three-dimensional channel flow with conjugate heat transfer is simulated. By the diagonal Cartesian method and 5-point Finite Analytic scheme, a grooved channel flow and flow in a casting bank at different Reynolds numbers are modeled. Simulations by both the diagonal Cartesian method and the traditional saw-tooth Cartesian method indicates the diagonal Cartesian method improves the modeling of flows, due to the more accurate approximation of complex boundaries. Heat transfer in two-dimensional finned compact heat exchanger is also studied. An improved heat exchanger is proposed based on the numerical prediction of heat transfer.

## Keywords

complex flow Cartesian diagonal finite analytic conjugate heat transfer## Nomenclature Alphabetic

*K*Thermal conductivity

*L*Characteristic length scale

*Nu*Nusselt number (ϱθ/ϱ

*n/w*)*P*Dimensionless Pressure (

*p/U*_{ref}^{2})*Pe*Peclet number (

*Re Pr*)*Pr*Prandtl number (

*v*/α)*Re*Reynolds number (

*U*_{ref}*L/v*)*S*Source term

*T*Temperature

*U*Velocity in the X direction

*V*Velocity in the Ydirection

*W*Velocity in the Z direction

*X*Spatial variable

*Y*Spatial variable

*Z*Spatial variable

*n*Normal direction from the wall

*p*Pressure

*u*Dimensionless velocity in the x direction (

*U/U*_{ref})*v*Dimensionless velocity in the y direction (

*V/U*_{ref})*w*Dimensionless velocity in the z direction (

*W/U*_{ref})*x*Dimensionless spatial variable (

*X/L*)*y*Dimensionless spatial variable (

*Y/L*)*z*Dimensionless spatial variable (

*Z/L*)

## Greek

- α
Thermal Diffusivity (

*K*/π*C*_{p})- Θ
Transport quantity

*v*Kinematic viscosity

- π
Density

## Subscript

*bc*Boundary condition

*f*Fluid

*p*Central node of finite analytic element

*ref*Reference

*s*Solid

*t*First order partial derivative in t

*w*Wall

*x*First order partial derivative in x

*xx*Second order partial derivative in x

*y*First order partial derivative in y

*yy*Second order partial derivative in y

*z*First order partial derivative in z

*zz*Second order partial derivative in z

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