Channel flow with viscous heating benchmark
Analysis Type: Convective heat transfer
this benchmark has been published by NAFEMS, it has not yet been implemented/tested using Feel++. 
1. Introduction
Channel flow is one of the basic flow arrangements that can be found in many engineering systems. In processes with highly viscous working fluids, shear may contribute importantly to the overall energy balance. Such viscous heating, often characterized by the Brinkman number, has to be taken into account in polymer processing as well as in the food process industry.
The thermally developing Poiseuille flow case has an educational value. It is a mathematically welldefined problem for which an analytical solution exists. For that reason, it is an integral part of many graduate courses in fluid mechanics.
2. Objectives
The case may represent a validation challenge as the momentum dissipation serves as a significant source of thermal energy. Although, the flow velocity profile is fully developed and steady, the imposed thermal boundary conditions cause a very pronounced thermal nearwall effect.
The model validation is based on the comparison of the simulated temperature profiles between two parallel plates with the analytical solution of the extended GraetzNusselt problem [1]. In addition, the flow mean temperature and the wall Nusselt number distribution along the channel wall are also used for the comparison.
Any effects of numerical diffusion will smoothen the temperature profile. Also, large diffusion constants (i.e. kinematic viscosity and thermal diffusivity) may impose stability restrictions on the size of the time step used in explicit CFD codes.
3. Geometry

Total length of the channel (H3) is 0.8 \(\mathrm{m}\); Channel halfwidth (V1=V2) is 0.01 \(\mathrm{m}\)

Domain height in zdirection is \(0.001 \mathrm{m}\) (although not important due to the case twodimensionality).
4. Case Definition
Four cases with a different pressure drop across the domain are investigated: \(\Delta p / L_{x}=4,40,400\) and \(4000 \mathrm{kPa} / \mathrm{m}\) The imposed pressure drop results in fully developed and steady flow. The inlet temperature \(\left(T_{\text {in}}\right)\) is set above the wall temperature \(\left(T_{w}\right)\) which cools the flow. It is the viscous heating associated with the fluid velocity gradient that raises the temperature in the flow.
5. Fluid Properties
The following fluid material properties that correspond to polymer processing are used for all investigated cases: \(\cdot \rho\) is density of \(500.0 \mathrm{kg} / \mathrm{m} 3\) \(\mu\) is dynamic viscosity of \(10.0 \mathrm{Pa} \mathrm{s}\) \(c_{p}\) is specific heat capacity of \(1000.0 \mathrm{J} / \mathrm{kgK}\) \(\lambda\) is thermal conductivity of \(1.0 \mathrm{W} / \mathrm{mK}\) The Prandtl number \(\left(P r=c_{p} \mu / \lambda\right)\) is 10,000 .
6. Initial Conditions
The solution of the extended GraetzNusselt problem can be simplified since the solution of the fully developed velocity field is known in advance.
In such case, a parabolic velocity profile [5.2] and zero pressure can be prescribed at the start of the simulation and left unchanged:
here \(b\) is the channel halfwidth.
7. Boundary Conditions
As the velocity field is known in advance, the solution of the momentum equation is not required. For that reason, only the energy equation boundary conditions are prescribed:

At the inlet boundary \(T_{i n}=30^{\circ} \mathrm{C}\)

At the top and bottom walls, the reduced temperature \(T_{w}=20^{\circ} \mathrm{C}\) is assigned;

For the vertical \(X\) Y surfaces, symmetry or equivalent conditions shall be used.
8. Output
The presented heat transfer problem and its solution are usually characterised with the following dimensionless numbers:

\(Pe = Re Pr\) is Peclet number;

\(Re=\rho u_{\max } b / \mu\) is Reynolds number;

\(Pr=c_{p} \mu / \lambda\) is Prandtl number;

\(Br=\mu u_{\max }^{2} / \lambda \Delta T\) is Brinkman number. The effect of axial thermal conduction can be neglected for large Peclet numbers (Pe \(\geq 100\) ), which leads to the following simplified differential equation:
where:
The exact solution of the nonhomogeneous differential equation can be found in the form of:

For validation purposes, the temperature profile \(\hat{T}(y)\) across the channel at \(x=0.4 m\) should be used. Four cases with a different imposed pressure drop lead to a different value of Br number, which impacts thermal conditions.

The flow mean temperature along the channel is defined as the mass weighted average temperature:
Its dimensionless form \(\hat{T}_{\text {mean}}=\left(T_{\text {mean}}T_{w}\right) / \Delta T\) should be calculated from the CFD results at intervals along the channel and compared with the analytical values. * The Nusselt number along the channel is defined as
where \(q w\) is the heat flux \((W / m 2)\) through the wall into the domain. The calculated Nusselt number at intervals along the channel shall be compared with the analytically obtained values.
Appendix A presents the analytically obtained values \(\hat{T}(y), \hat{T}_{\operatorname{mean}}(x)\) and \(N u(x)\) in a table form.
Standard deviations can be used to quantify the level of agreement between the analytical and the CFD simulation results.
9. References
1 R.K. Shah and A.L. London, Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data: Laminar Flow Forced Convection in Ducts, Academic Press, 1978, p170 2 H. Schlichting, Boundary Layer Theory, McGrawHill, 7 th \(\mathrm{Ed}, 1979\) p. 85,277 \& 281