# Modeling Turbulent Heat Flux Distribution

### Key Takeaways

• Turbulent heat transfer mechanisms are extensively used in engineering and technological processes.

• Turbulent thermal convection problems are often addressed using the paradigm system, or the Rayleigh Benard (RB) convection system.

• It is important to model turbulent heat flux distribution and behavior to further enhance heat exchange efficiency and performance.

Most fluid flow follows turbulent characteristics. When describing the turbulent flow mathematically using Navier-Stokes equations, a few limitations are usually encountered. Finding a unique solution for the Navier-Stokes equations that satisfies initial conditions and continues to remain valid in turbulent fluid flow is a common challenge engineers face.

Atmospheric turbulence is one example where the chaotic nature of turbulent heat flux makes solving the governing equations mathematically challenging. In the atmosphere, turbulent heat fluxes, namely sensible heat flux and latent heat flux, are present. Sensible and latent turbulent heat fluxes play an important role in the transport of energy back to the atmosphere. Similarly to atmospheric turbulence, turbulent heat flux is present in several engineering and technological systems. We will explore turbulent heat flux in this article.

## Atmospheric Turbulent Heat Flux

The energy received by Earth’s surface in the form of radiation is transported back to the atmosphere due to turbulent heat fluxes. The turbulent heat exchange in the atmosphere occurs over scales of motion in the range of millimeters to kilometers. There are two turbulent heat fluxes, sensible heat flux and latent heat flux, in the atmosphere that cause energy transport. Sensible heat flux is responsible for atmospheric heating up to around 100m during the daytime.

While turbulence occurs naturally in the atmosphere and oceans, humans have also included turbulence in some engineering and technological processes. Let’s take a look at turbulent thermal convection and how turbulent heat fluxes are measured in the upcoming section.

## The Relationship Between the Rayleigh Number, Prandtl Number, and Nusselt Number and Turbulent Thermal Convection

Turbulent thermal convection is utilized in engineering technology and industrial systems for heat transfer and mixing. Turbulent thermal convection problems are often addressed using the paradigm system, or the Rayleigh Benard (RB) convection system. The following dimensionless numbers have great relevance in describing turbulent heat transfer and turbulent thermal convection.

1. Rayleigh Number - A dimensionless quantity that describes the laminar or turbulent nature of natural convection heat transfer. The Rayleigh number is related to the Grashof number and the Prandtl number as follows:

## Rax = Grx * Pr

1. Prandtl Number - The Prandtl number is expressed as the ratio of the momentum diffusivity to thermal diffusivity. It gives the similarities between the turbulent momentum exchange and turbulent transfer capacity in a fluid. The Prandtl number is an intrinsic property of the fluid.

2. Nusselt Number - Convection heat transfer occurring at the fluid surface can be measured from the Nusselt number. The Nusselt number can be expressed as the unitless temperature gradient at the fluid surface.

In the RB convection system, the Prandtl and Rayleigh numbers determine flow dynamics. The turbulent heat flux in such systems is measured in terms of the Nusselt number, and its dependency on Rayleigh and Prandtl numbers is given by prefactors obtained from experimental data.

## Modeling the Turbulent Heat Flux Distribution

Turbulent flow and heat transfer are omnipresent in industrial processes. For example, in heat exchanger applications, turbulent flow and thermal transport are utilized. In such systems, the nature or texture of the wall influences the efficiency of the process and the heat flux distribution.

When the turbulent flow is confined to a solid surface, boundary layers develop in the vicinity of the walls. The velocity boundary layer has zero value near the wall surface and attains a substantial value in the core of the flow. Similarly, the temperature varies from hot (bottom) to cold (top) temperature through intermediate temperatures at the core of the flow.

The velocity and temperature gradients formed by the boundary layer in RB turbulent convection influence the momentum distribution and heat flux distribution. It is essential to model turbulent heat flux distribution and behavior to further enhance heat exchange efficiency and performance.

The turbulent heat flux transport equation is the mathematical base for modeling heat exchanger systems. Depending on the flow characteristics, there is usually a convective term, diffusion term, and pressure-temperature gradient term present in the turbulent heat flux transport equation. By accurately modeling the turbulent heat transport, it is possible to predict the mean temperature distribution and turbulent heat flux component distribution in the system under consideration. The overall accuracy of thermal transport systems can be enhanced by acquiring knowledge from turbulent heat flux models.

### The Dynamics of Turbulent Heat Flux

The dynamics associated with turbulent heat flux distribution and behavior require the modeling of turbulent thermal transport systems. Cadence’s CFD tools can assist you with modeling turbulent heat flux distribution in dynamic fluid flow systems with an industry-leading meshing approach, robust solvers, and post-processing capabilities.