Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5907856, 7 pages

https://doi.org/10.1155/2017/5907856

## Stochastic Analysis of Natural Convection in Vertical Channels with Random Wall Temperature

Department of Mechanical Systems Engineering, National Institute of Technology, Asahikawa College, 2-2-1-6 Shunkodai, Asahikawa 071-8142, Japan

Correspondence should be addressed to Ryoichi Chiba

Received 31 May 2017; Accepted 17 July 2017; Published 13 August 2017

Academic Editor: Sergey A. Suslov

Copyright © 2017 Ryoichi Chiba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study attempts to derive the statistics of temperature and velocity fields of laminar natural convection in a heated vertical channel with random wall temperature. The wall temperature is expressed as a random function with respect to time, or a random process. First, analytical solutions of the transient temperature and flow velocity fields for an arbitrary temporal variation in the channel wall temperature are obtained by the integral transform and convolution theorem. Second, the autocorrelations of the temperature and velocity are formed from the solutions, assuming a stationarity in time. The mean square values of temperature and velocity are computed under the condition that the fluctuation in the channel wall temperature can be considered as white noise or a stationary Markov process. Numerical results demonstrate that a decrease in the Prandtl number or an increase in the correlation time of the random process increases the level of mean square velocity but does not change its spatial distribution tendency, which is a bell-shaped profile with a peak at a certain horizontal distance from the channel wall. The peak position is not substantially affected by the Prandtl number or the correlation time.

#### 1. Introduction

Recently, as reliability gains increasing importance in the design phase of thermal systems, conventional deterministic heat transfer analysis alone is not sufficient; analysis that considers uncertainties included in the systems themselves and/or thermal environments (in other words, heating/cooling conditions) is required. In general, accurately predicting the thermal or mechanical loads acting on components in high-temperature apparatus is very difficult. This alludes to the fact that many uncertain factors exist in the design of such apparatus [1]. In this situation, the temperature fields in solid objects and/or working fluid should be estimated stochastically.

There has been an interest in analysing and quantifying the effects of uncontrollable random factors on the fluid flow and heat transfer performances of thermal systems. To estimate the effects quantitatively, probabilistic methods have been applied to heat transfer problems. Existing literature about the application of the methods to heat conduction problems is summarised in review articles [2–4], and therefore we do not enter further into the subject here. On the other hand, focusing on the application to convective heat transfer problems, one can find early studies of stochastic forced convection with heat source and initial and boundary conditions being white noises [5, 6]. Subsequently, other researchers investigated the effects of random temperature and velocity of a moving wall on the temperature field in a Couette flow [7], the effects of boundary temperature and boundary topology modelled by random fields on the temperature and velocity fields of natural convection in a square domain [8, 9], and the effects of boundary temperature, which was assumed to be a random field, on the Nusselt numbers of mixed convection in a horizontal channel [10].

However, the effects of randomly fluctuating the temperature of a bounding wall on the fluid flow and thermal performance of natural convection in channels have not been investigated. The temperature fluctuations at the wall are not necessarily zero in real situations [11], which are caused by various natural noises. In addition, the effects of wall temperature fluctuations on the evolution of the fluid flow and heat transfer are significant in internal natural convection [12].

In this paper, we address the stochastic natural convection problem of viscous fluid in an infinite vertical channel with temporally random wall temperature. Under the assumption that the wall temperature is a stationary random process, the autocorrelation functions are analytically derived for the fluid temperature and flow velocity. Numerical calculations are performed under the condition that the wall temperature is expressed either as white noise or as a stationary Markov process. We quantify the effects of the Prandtl number and the correlation time of the wall temperature fluctuation on the mean square values (i.e., second moments) of the fluid temperature and velocity. This stochastic problem may also be solved numerically by applying a variance propagation algorithm based on the finite element formulation [13]. However, our analytical solutions to the second moments are mathematically sounder.

#### 2. Analysis of Natural Convection Problem

Let us consider the transient laminar natural convection of a viscous incompressible fluid in an infinite vertical channel, as shown schematically in Figure 1. Two flat plates having infinite width and length (i.e., doubly infinite plates) are placed vertically with a separation distance of in the fluid. Initially, the fluid is quiescent at uniform temperature , and the vertical plates are maintained at the same temperature . It is assumed that the temperature of the plate located at fluctuates randomly for time whereas the other plate is kept constant at . The physical properties of the fluid are constant except for its density; density variation is considered only in the buoyancy term, which is a linear function of temperature (i.e., the usual Boussinesq approximation). Viscous dissipation effects are neglected.