**Instructors:** Georgios Sylaios, Konstantinos Moutsopoulos

**Course Code:** 15ΗΕ1Ν – Κ2

**Semester:** 8th

**Weekly teaching hours: **6

**ECTS credits:** 5

**Prerequisites: **Physical Oceanography, Management of Inland and Coastal Systems, Fluid Mechanics, Applied and Groundwater hydraulics

**Course offered to Erasmus students: **No

**Course URL: **https://eclass.duth.gr/courses/TMC358/

https://eclass.duth.gr/modules/document/ ?course=TMC137

**Learning Outcomes: : **

**Α) Knowledge-based**

- Student introduction to the concepts of environmental fluid mechanics,
- Understanding the pollutant transport and mixing processes in one and two-dimensional flows,
- Understanding the basic equations describing the processes of pollutant transport and mixing in rivers, lakes, aquifers and coastal seas,
- Understanding the mechanisms of dispersion from submarine diffusers,
- Comprehending the numerical errors and the criteria for the assessment of numerical schemes.ς.

**Β) Skils/Competences acquired**

- Capacity to solve numerically the environmental fluid mechanics equations,.
- Ability to configure environmental flow models,
- Capacity to design coastal submarine diffusers,
- Capacity to select the appropriate numerical model and numerical scheme to solve a groundwater flow problem.

**General Skills: **

ICT use, Decision-making, Project design and management,critical thinking, autonomous work.

**Course Content:**

This course introduces the student to the principles, the theoretical background and the main equations of environmental fluid mechanics, as well as its application in the natural aquatic environment (streams, torrents, rivers, lakes and reservoirs, coastal sea, aquifers). The first part of the course is devoted to the presentation of the main theoretical equations of environmental fluid mechanics and the modeling principles of environmental flows. The methods of solving the diffusion equation for the one-dimensional and two-dimensional problems using explicit and implicit numerical schemes are presented. Special attention is given to the methods of Gauss-Seidel, Jacobi and ADI (Alternate Direction Implicit). Similarly, the simple advection problems in one and two-dimensions are explained. The problems of scheme convergence and stability and the different types of numerical errors are presented. The equations of Navier-Stokes, continuity, integrated in depth continuity and momentum, laterally integrated continuity and momentum, and the equations of advection-diffusion are shown. Further, the fluid mechanics of groundwater flows are presented. The empirical methods for the estimation of turbulent viscosity and turbulent diffusivity coefficients are presented. The theoretical background and the practical applications of k-epsilon turbidity model are shown. At the second part of the course, we are focusing into applied environmental flows and processes. More specifically, the processes of advection – diffusion in 1-dimensional flows (streams, rivers), the processes of lake stratification – mixing and their effect on pollutants’ distribution, the two-dimensional flows in lakes, estuaries and coastal seas, and the three-dimensional processes of open seas are explained. The course presents the study of plumes and buoyancy jets from submarine diffusers, and focuses on the methods to optimize their design characteristics. The basic models of groundwater flows are presented for different flow types (Darcy flows, non-Darcy flows, flows in aquifers under pressure, and aquifers with double porosity, etc. Finally, specific examples in modeling and management of water resources are shown.

**Theoretical Lectures:**

1. Diffusion and advection of contaminants. Solutions using analytical and numerical methods

2. Transformation of the Navier-Stokes equation for surface water systems. Depth-integrated formulation using the continuity equation. The Saint-Venant equations and their solutions

3. One dimensional advection-diffusion of contaminants. Application in channels, torrents and lakes.

4. One-dimensional simulation of stratification / destratification of the water column. Application to lakes and reservoirs. The PHYTO model

5. Two-dimensional simulation of the hydrodynamic dispersion in seas in the near-coast area.

6. Underwater jets and plumes. Design of wastewater diffusors

7. Presentation of the Finite Difference Method. Approximation of the derivatives using the Taylor series. Accuracy of the derivatives approximation. Algebraic expressions for the first and second derivatives. Examples.

8. Explicit numerical schemes. The FTCS (Forward in Time Central in Space) method. The DuFort-Frankel method. Application to the solution of pure diffusion.

9. Solutions of Partial Differential Equations using implicit methods. Solution of the resulting systems of algebraic equations using direct methods. The method Gauss. The Method LU. The method TDMA (Tri Diagonal Matrix Algorithm)

10. Solutions of Partial Differential Equations using implicit methods. Solution of the resulting systems of algebraic equations using indirect methods. The Gauss-Seidel method. Under-relaxation and over-relaxations. The ADI (Alternate Direction Implicit) method.

11. The Finite Volume Method..

12. The k-ε model for the simulation of turbulent flows. The Random Walk Method. Applications to the solution of the advection- diffusion equation.

13. Solution of Partial Differential Equations related to Groundwater Hydraulics; Solutions to the Boussinesq equation, to the Double Porosity equations and to the Forchheimer equation.

**Exercises/ Practicals:**

1. Numerical solution of diffusion equation in one-dimensional system,

2. Numerical solution of advection equation in one-dimensional system,

3. Numerical solution of advection-diffusion equation in one-dimensional system,

4. Dimensional analysis of Navier-Stokes equations,

5. Numerical solution of St Venant equations along a river,

6. Numerical solution of pollutant dispersion along rivers,

7. Submarine dispersion of pollutants from plumes and jets,

8. Examples with the MODFLOW software,

9. Numerical solution of groundwater flow problems.

**Suggested Bibliography:**

1. «Environmental Computational Fluid Mechanics», Sylaios Georgios & Moutsopoulos Konstantinos, 2015, KALLIPOS e-book.

2. «Environmental Fluid Mechanics», Dimitriou I., 448 p.

3. «Environmental Models», Scnoor, J., 768 p.

4. «Computational Fluid Mechanics», Markatos and Asimakopoulos, 206 p.

- Μαρκάτος Ν., Δ. Ασημακόπουλος 1995. Υπολογιστική ρευστοδυναμική, Εκδ. Παπασωτηρίου, σ. 206