INFORMAZIONI SU

Computational and Fluvial Hydraulics (Idraulica computazionale e fluviale)

Programma dell'insegnamento di Computational and Fluvial Hydraulics (Idraulica computazionale e fluviale) - cdl magistrale in Ingegneria Civile - mutua dal cdl magistrale in Ingegneria per l'Ambiente e il Territorio

Teacher

prof. aggr. Silvia BOSA

Credits

6 CFU

Objectives

The course comprehend both theoretic and practical applications. The objectives of the course are to teach:
- the basic notions of fluvial hydraulics;
- the fundamental of numerical analysis and the main techniques for the solution of differential equations, that describe the principles of fluvial hydraulics;
- how to build easy numerical models by means of MatLab language.


Contents

RECALL OF HYDRAULICS PRINCIPIA: Dimensional analysis. 1D flow. Mass and momentum conservation. Laminar and turbulent flow. Boussinesq approximation. Differential equation of motion for inviscid fluids, laminar and turbulent flow. Flow resistance.(4 hours)
RHEOLOGICAL MODELS: Elastic and viscoelastic model. Non-Newtonian fluids. Bingham fluid. Pseudoplastic and dilatant fluid. Viscoplastic Modello di fluido pseudoplastico e dilatante. Hershel-Bulkley fluid. (2 hours)
1D SHALLOW WATER EQUATIONS: 1D continuity equation. 1D momentum equation. Shallow water equations (De saint Venant equations for prismatic channels). Hydrostatic pressure distribution. Weak bottom slope. Vectorial form of De Saint Venant equations. Primitive and conservative variables. Initial and boundary conditions. Mathematical and physical conservativity of De Saint Venant equations.(3 hours)
2DH SHALLOW WATER EQUATIONS: 2DH continuity equation. 2DH momentum equation. Shallow water equations (SWE). Vectorial form of SWE. Euler, Navier-Stokes and Reynolds equations in shallow water. SWE with primitive and conservative variables. Initial and boundary conditions. Mathematical and physical conservativity of SWE. (3 hours)
FLOW WAVES: Kinematic model. Diffusive wave model. Flow rate. Flow wave propagation and flooding: regulation and some considerations on the applicability of 1D and 2DH models. (2 hours)
SEDIMENT TRANSPORT: Sediment properties. Size of the sediment. Dimension, form, density, settling velocity of the sediment. Granulometry, porosity and angle of repose. Initiation of motion. Critical shear stress for bed load and sospende load transport. Bedforms. Flow resistance in mobile river bed. Sediment transport rate. Evaluation of bed load. Mayer-Peter and Müller formula. Van Rijn formula. Evaluation of suspended load. Rouse model. Van Rijn formula. Practical criterions. Bed material load evaluation. Engelund & Hansen and Ackers & White formulae. Exner equation. 1D and 2DH models to study sediment transport. (6 hours)
MUD AND DEBRIS FLOW: Introduction. Mechanics of debris flow. Uniform shear flow of particles moving as a passive tracer. Flow resistance. Debris flow classification. Single-phase models. Frictional model. Herschel-Bulkley model. Bagnold and O’Brien model. 2DH models. (3 hours)
ERROR ANALYSIS: Absolute and relative error. Source of errors in data. Computer representation of numbers. Error propagation. Main vectorial norms. (2 hours)
MATRIX ALGEBRA: Recall of symbology and matrix algebra. (1 hour)
SOLUTION OF LINEAR SYSTEM OF EQUATIONS: Solution of a linear system. Direct and iterative methods. Direct methods. Gaussian elimination. Partial and complete Pivoting. LU decomposition. Iterative methods. Jacobi and Gauss – Seidel iteration. (2 hours)
SOLUTION OF NONLINEAR EQUATIONS: Bracketing methods (bisection method). Open methods (Newton-Raphson method, secant method, regula falsi). Fixed-point iteration. Newton-Raphson method for systems of nonlinear equations. (2 hours)
DATA APPROXIMATION: Polynomial interpolation. Lagrange polynomials. piecewise polynomials and spline. Least squares approximation. (2 hours)
NUMERICAL INTEGRATION: Quadrature formulae based on Polynomial interpolation. Newton–Cotes rules (trapezoid rule, Simpson's rule, Newton-Cotes coefficients). (2 hours)
ORDINARY DIFFERENTIAL EQUATIONS (ODE): Cauchy problem. Finite difference formulae (Euler and backward Euler method, midpoint method, Crank–Nicolson method, Heun method, modified Euler method). Error analysis in one step methods (global error, local error, error propagation; convergence, consistency, and stability; accuracy order). Implicit and predictor-corrector methods. (2 hours)
PARTIAL DIFFERENTIAL EQUATIONS (PDE): Hyperbolic, parabolic and elliptic equations. Physical role of this classification. Boundary conditions (Dirichlet, Neumann, mixed). Initial conditions. Characteristic curves and classification through characteristics (3 hours)
FINITE DIFFERENCE METHOD: Discretization of differential operators. Example for PDE. First and second order schemes (forward and centered finite difference). Implicit and explicit schemes, upwind and centered schemes. Higher order schemes. Mentions of variable grid-size. Initial and boundary conditions. Convergence, consistency, and stability. Consistency analysis. Von Neumann stability analysis. Courant Friedrichs Lewy condition. Convergence order. Accuracy. (4 hours)
CHARACTERISTIC METHOD: Hyperbolicity of De Saint Venant equations. Characteristic curves. Riemann invariants. Method of characteristics (subcritical and supercritical flow, solution through the method of characteristics, initial and boundary conditions, curved and straight characteristics, range of influence, zone of determinacy, domain of dependence). Dam break on dry and wet bed. Riemann problem. Ritter and Stoker problems. General solution. (2 hours)
FINITE VOLUME METHOD (FVM) APPLIED TO 1D FLOW: Spatial discretization. Finite volume method. Temporal integration. Intercell flux evaluation. Courant-Friedrichs-Lewy condition. Godunov method. Exact Godunov solver. Upwind and centered approximate Riemann solvers (centered Godunov, Lax-Wendroff, Lax-Friedrichs, FORCE, HLL). Source term. Second order schemes in space (linear variable reconstruction, limiters). Second order schemes in time. Initial and boundary conditions. (3 hours)
FINITE VOLUME METHOD APPLIED TO 2DH FLOW: Spatial discretization. x-splitting method. FVM in a 2DH domain. Solution of 1D augmented Riemann problem. Intercell flux evaluation. Exact Godunov solver. HLLC approximate Riemann solver. Temporal integration. Source term. Courant-Friedrichs-Lewy condition. Second order schemes in space and time. Initial and boundary conditions. (2 hours)
MATLAB: Basic instructions. How to build a program in Matlab. Applications: find the root of a real-valued function; data approximation; numerical integration; finite volume model for the solution of 1D SWE. (10 hours)

Type of exam:

Oral