This thesis develops data-driven flow reconstruction methods to reconstruct the velocity of plane Couette flow from wall temperature. We performed a Direct Numerical Simulation (DNS) for a heated plane Couette flow with imposed flux boundary condition at the bottom wall to create a data-set. Due to the the imposed flux the temperature at bottom wall is free and wall temperature patterns can develop. The focus of this thesis is on the investigations of the strong correlation between the flow velocity and the wall temperature. We analyse their joint probability density function and cross variance spectrum to develop a spectral linear regression model. This model successfully reconstructs wall shear stress from wall temperature except possibly at peaks. To reconstruct flow velocity from wall temperature, we apply flow decomposition modes such as the Proper Orthogonal Decomposition (POD) modes \cite{Holmes2012ProperDecomposition}. We design test problems to develop a framework to reconstruct \emph{gappy} fields with missing information. In this framework, we prescribe suitable regularisation for the under-determined \emph{gappy} fields. We also develop a decomposition method - the subdomain POD method which divides a physical domain into a number of subdomains and then applies the POD method in each subdomain individually. This subdomain POD are locally optimised and inherits properties of the POD modes. In both cases, namely the POD and the subdomain POD method, the reconstructions are found to be in good agreement with the flow velocity obtained form the DNS. To develop data-driven methods with imposed physical constraints, we propose a linear dynamical model based on Orr-Sommerfeld-Squire \cite{Kim2007,Murray2006} system and the scalar transport equation. This model successfully reconstruct some of the key flow structures at $z^+=35$.