Studying free surface flows generated by a moving body is relevant to various fluid-structure interaction applications. Such investigations are crucial for fields such as ship and submarine design, where understanding fluid dynamics and wave motion is essential. Although numerous problems involve bodies moving at constant speed in water, the phenomenon of wave generation by submerged moving structural bodies remains a longstanding and challenging problem. Despite decades of investigation [1,3], progress in this area has been limited, with few significant advances in recent years. At the same time, the rapid growth in high-performance computing in the past decades enables the use of state-of-the-art computational fluid dynamics methods of increasing fidelity for predicting flows and continues to impact engineering design processes. We present a new high-order numerical solver capable of finding steady-state solutions to the incompressible Navier-Stokes equations subject to various flow conditions, different submerged structures, and a free surface. With this objective, the mathematical task is twofold: determine i) the final shape of the free surface and ii) the flow quantities, including velocity and pressure. The governing equations are handled within the open-source parallel computational framework, Firedrake [2], where they are solved considering a Galerkin-based numerical discretization. In particular, the weak form of the problem is addressed using a spectral element method (SEM) [4]. Since the free surface dynamically adapts to the pressure and velocity fields of the flow, an iterative approach is employed to reach the steady-state profile through dynamic adjustments of the free surface boundary. The proposed spectral element solver is first validated by simulating the flow around a stationary cylinder. Subsequently, the domain is extended to include a submerged NACA airfoil traveling at a constant velocity, generating a steady wave on the free surface. In addition, these results are qualitatively compared with those reported in [1].