Eddy thermal diffusivity model and mean temperature profiles in turbulent vertical convection

This paper proposes a space-dependent eddy thermal diffusivity model to derive analytical mean temperature profiles for turbulent vertical natural convection, revealing universal scaling functions in inner and outer regions that align well with direct numerical simulation data.

The Gist

The Big Picture: A Hot Wall and a Cold Wall

Imagine a room with two giant, infinite walls standing side-by-side. One wall is boiling hot (like a radiator), and the other is freezing cold (like a block of ice). The air (or water) trapped between them wants to move. The hot fluid near the warm wall rises, and the cold fluid near the cold wall sinks. This creates a swirling, chaotic dance known as turbulent convection.

Scientists have known the rules for laminar flow (smooth, orderly movement) for a long time. But when the flow gets turbulent (chaotic, like whitewater rapids), it's incredibly hard to predict exactly how the temperature changes as you move from the hot wall to the cold wall.

This paper is about building a better map to predict that temperature change.

The Problem: The "Black Box" of Turbulence

In smooth flow, you can calculate exactly how heat moves. But in turbulent flow, the fluid is churning so wildly that heat is carried by giant, invisible eddies (swirls) rather than just slowly diffusing.

To predict the temperature, scientists usually use a concept called Eddy Thermal Diffusivity. Think of this as a "chaos factor." It measures how much the swirling motion helps heat spread out.

• The old way: Previous models tried to guess this "chaos factor" using simple math (like a straight line or a simple curve).
• The problem: These old guesses worked okay for air, but they fell apart when the fluid was thicker (like oil or water) or when the temperature difference was huge. They couldn't explain the data from super-computer simulations.

The Solution: A Three-Layer Cake

The authors, Ho Yin Ng and Emily Ching, realized that the space between the walls isn't uniform. It's more like a three-layer cake, where each layer behaves differently:

1. The Crust (The Inner Region): Right next to the hot and cold walls, the fluid is stuck to the surface (it can't slide). Here, the "chaos" is low. The heat moves mostly by conduction (like a spoon getting hot in soup). The "chaos factor" here grows slowly, like a cube of a number.
2. The Filling (The Middle Region): As you move away from the wall, the turbulence kicks in. The fluid starts swirling wildly. Here, the "chaos factor" changes in a straight-line fashion.
3. The Center (The Outer Region): Right in the middle of the gap, the flow is most turbulent. The heat is being mixed furiously. Here, the "chaos factor" peaks and then drops off symmetrically, like a hill.

The Analogy: Imagine walking from a quiet library (the wall) into a mosh pit (the center).

• Near the wall, people are just standing still (low chaos).
• In the middle, people are shoving and dancing wildly (high chaos).
• The authors created a mathematical rule that describes exactly how "shoving" changes as you walk from the library to the center.

The Discovery: Two Universal Rules

Using this "Three-Layer Cake" model, the authors derived a formula for the temperature. They found something beautiful:

No matter how thick the fluid is (Prandtl number) or how hot the walls are (Rayleigh number), the temperature profile always collapses into two universal shapes:

1. The Wall Shape: A specific curve that describes how temperature drops right next to the wall.
2. The Center Shape: A different specific curve that describes how temperature behaves in the middle.

It's like saying that whether you are driving a Ferrari or a tractor, or whether you are driving on a rainy day or a sunny day, the shape of the road curve near the exit ramp is always the same, and the shape of the straight highway in the middle is also always the same. You just need to know how to "scale" your speed and position to fit the map.

Why This Matters

• Better Predictions: Their model fits the data from super-computer simulations (Direct Numerical Simulation) much better than previous theories. It works for everything from air (thin) to heavy oils (thick).
• Real World Applications: This helps engineers design better ventilation systems for buildings, predict how ice melts in the ocean near vertical ice shelves, and improve heat exchangers in power plants.
• Debunking Old Myths: They proved that some older theories (which suggested temperature drops in a specific "inverse cube root" way or follows a logarithmic line everywhere) were actually wrong for many fluids. The "Three-Layer" approach is the correct way to see it.

In a Nutshell

The authors built a new, more accurate "rulebook" for how heat moves in a turbulent fluid between two walls. Instead of using one simple rule for the whole space, they realized the space has three distinct zones. By treating each zone with the right math, they found that nature follows two simple, universal patterns for temperature, regardless of the fluid's thickness or the heat intensity. It's a clearer, more accurate map for understanding the chaotic dance of heat.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding turbulent vertical natural convection in a fluid confined between two infinite vertical walls maintained at different temperatures ($T_h$ and $T_c$). While laminar vertical convection is well-understood via boundary-layer analysis, a complete theoretical framework for the turbulent regime remains elusive.

Key physical quantities of interest include the Nusselt number ($Nu$, representing heat flux), wall shear stress, and the mean temperature and velocity profiles. Previous theoretical attempts (e.g., George & Capp 1979; H¨olling & Herwig 2005) proposed scaling laws for these profiles (such as inverse cubic-root or logarithmic dependencies) but often failed to hold universally across different Prandtl numbers ($Pr$) or violated boundary conditions. The authors aim to derive analytical mean temperature profiles that are valid for a wide range of $Pr$ and Rayleigh numbers ($Ra$) by proposing a new closure model.

2. Methodology

The authors employ a theoretical approach based on the Reynolds-averaged Navier-Stokes (RANS) equations under the Oberbeck-Boussinesq approximation.

• Governing Equations: The study focuses on the mean temperature equation (Eq. 2.5):
  $$\frac{d}{dx}\overline{u'T'} = \kappa \frac{d^2\overline{T}}{dx^2}$$
  where $\overline{u'T'}$ is the turbulent heat flux.
• Closure Model: To close the system, the authors propose a space-dependent eddy thermal diffusivity model, $K(x)$, defined by the relation $\overline{u'T'} = -K(x) \frac{d\overline{T}}{dx}$.
• Three-Layer Structure: Recognizing that molecular diffusivity dominates near the walls while turbulence dominates near the centerline, the domain is divided into three regions:
  1. Inner Region ($0 \le y \le y_1$): Near the wall. $K(x)$ is modeled as a cubic function ($Ay^3$) to satisfy the no-slip and isothermal boundary conditions (where turbulent flux and its derivatives vanish at the wall).
  2. Intermediate Region ($y_1 < y < y_2$): A linear function connects the inner and outer regions.
  3. Outer Region ($y_2 \le y \le 1/2$): Near the centerline. $K(x)$ is modeled as a symmetric quadratic function peaking at the centerline ($x=H/2$).
• Parameters: The model is governed by two independent parameters, $A$ and $C_m$, which determine the characteristic velocities for heat transfer in the inner ($V_i$) and outer ($V_o$) regions, respectively.
• Validation: The analytical results derived from this model are validated against Direct Numerical Simulation (DNS) data from Howland et al. (2022) covering $1 \le Pr \le 100$ and $10^6 \le Ra \le 10^9$.

3. Key Contributions

• Novel Eddy Diffusivity Model: The paper introduces a three-layer eddy thermal diffusivity model that rigorously satisfies boundary conditions at the walls and symmetry at the centerline, overcoming limitations of previous single-layer or power-law models.
• Universal Scaling Functions: The authors derive two distinct, universal scaling functions for the mean temperature profile:
  • Inner Scaling ($F_i$): Describes the profile near the walls.
  • Outer Scaling ($F_o$): Describes the profile near the centerline.
• Resolution of Prandtl Number Dependence: Unlike previous theories that struggled to collapse data for different $Pr$, this model incorporates the Prandtl number explicitly into the characteristic scales, allowing for a universal description across a wide range of fluids (from liquid metals to oils).
• Analytical Derivation: The study provides closed-form analytical expressions for the mean temperature profile and the Nusselt number, avoiding the need for purely numerical fitting.

4. Key Results

• Mean Temperature Profiles:
  • In the inner region, the rescaled temperature $(T_h - T)/T_i$ collapses onto a single universal curve described by the function $F_i(z)$, which involves logarithmic and arctangent terms. This function deviates from a simple linear profile, indicating significant turbulent heat flux even within the inner layer.
  • In the outer region, the rescaled temperature $(T - T_m)/T_o$ collapses onto a universal curve described by $F_o(z)$, which follows a logarithmic form (specifically $\propto \log((1-z)/z)$).
• Parameter Dependencies:
  • The parameters $A$ and $C_m$ are found to scale with $Ra$ and $Pr$. Specifically, $A \propto Ra$ and $C_m \propto Ra^{4/9}$ in the high-$Ra$ limit.
  • The model successfully predicts the high-$Ra$ scaling law $Nu \sim Ra^{1/3}$, consistent with previous theoretical and experimental findings.
• Comparison with DNS:
  • The model shows excellent agreement with DNS data for $Nu$ and mean temperature profiles across all tested $Pr$ and $Ra$.
  • The relative error in predicting $Nu$ is significantly lower (typically < 10%) compared to models that assume a single power-law diffusivity throughout the domain (which showed errors up to 47%).
• Critique of Previous Models:
  • The paper demonstrates that the "inverse cubic-root" profile proposed by George & Capp (1979) fails to collapse data for high $Pr$.
  • The "logarithmic" profile proposed by H¨olling & Herwig (2005) is shown to be inconsistent with the data when tested against the derived $Nu$-$Ra$-$Pr$ relationship.
  • The model by Li et al. (2023) is shown to violate boundary conditions (non-vanishing derivatives at the wall) and fails to collapse the outer region data onto a straight line as predicted.

5. Significance

This work provides a robust theoretical framework for turbulent vertical convection, a fundamental problem in fluid dynamics with applications in building ventilation, geophysical flows, and polar ocean-ice interactions. By establishing universal scaling functions that depend on physically motivated characteristic velocities, the authors resolve long-standing discrepancies in the literature regarding the shape of mean temperature profiles across different fluid properties. The model's ability to accurately predict heat transfer rates ($Nu$) and temperature profiles for a wide range of $Pr$ and $Ra$ makes it a valuable tool for engineering design and further theoretical development in turbulent convection.