Int. J. Mech. Eng. Autom. Volume 2, Number 11, 2015, pp. 514-524 Received: September 29, 2015; Published: November 25, 2015 International Journal of Mechanical Engineering and Automation An Efficient 3D, Implicit Scheme for Free-Surface Flow Calculation with Shockwaves Athanasios J. Klonidis and Johannes V. Soulis Dept. of Civil Engineering, Democritus University of Thrace, Xanthi 67132, Greece Corresponding author: Athanasios J. Klonidis (klonidis@gmail.com) Abstract: This paper refers to the development of a moving grid implicit finite-volume numerical scheme capable of simulating 3D, steady, free-surface flows in irregular geometry channels with shockwave presence. The Navier-Stokes equations are used together with pseudo-compressibility technique to allow direct calculation of the pressure from the continuity equation. The position of the free-surface is determined by applying a moving boundary condition through the inclusion of the two-dimensional depth-averaged mass continuity equation. To improve the stability and accuracy of the model a new technique is introduced based on the use of two nested iteration steps. All of the mentioned equations are transformed into non-orthogonal body-fitted coordinate system to enable accurate representation of irregular geometries. The resulting numerical model is used to simulate super-critical free-surface flows including discontinuous flow featured with shock waves. The comparisons are satisfactory. Keywords: Finite-volume scheme, implicit numerical scheme, non-orthogonal boundary fitted coordinates, shock wave, steady flow, three-dimensional free-surface flow. Nomenclature A = ∂F/∂Q Jacobian matrix (dimensionless) B = ∂G/∂Q Jacobian matrix (dimensionless) C = ∂H/∂Q Jacobian matrix (dimensionless) g Gravity acceleration (m·s -2 ) h Water depth (m) I Identity matrix (dimensionless) J Determinant of transformation matrix (dimensionless) k Index of current internal iteration step (dimensionless) k+1 Index of next internal iteration step (dimensionless) n Index of current external iteration step (dimensionless) n+1 Index of next external iteration step (dimensionless) P Pressure (N·m -2 ) Q Flow discharge (m 3 ·s -1 ) S fx , S fy friction slopes along x- and y-direction (dimensionless) t Time (s) u * Friction velocity (m·s -1 ) v x , v y , v z Velocity components in axial (x) tangential (y) and vertical direction (m·s -1 ) v ξ , v Ș , v ȗ Local (computational) velocity components along ξ-, Ș- and ȗ-direction (m·s -1 ) x v , y v Depth averaged velocity components in axial (x) and tangential (y) direction (m·s -1 ) ξ v , η v Local (computational) mean velocity components along ξ-, and Ș-direction (m·s -1 ) v τ Turbulent kinematic viscosity (m 2 ·s -1 ) x, y, z Cartesian coordinates (m) β Pseudo-compressibility factor (dimensionless) δ Central difference (dimensionless) Δt Time step (s) Δx, Δy, Δz Spatial steps in Cartesian coordinate system (m)