# HYDRODYNAMIC STABILITY AND BOUNDARY LAYERS - 2025/6

Module code: MATM070

## Module Overview

This module is an introduction to the concepts of fluid stability in parallel flows, such as channel flows, and non-parallel flows such as boundary layer flows. By the end of the course students should understand the concept of hydrodynamic stability in the context of simple parallel flows. They should be able to derive the governing stability differential equations and analyse the stability properties of a range of both inviscid and viscous flows.

### Module provider

Mathematics & Physics

GODOLPHIN Janet (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Semester 2

None.

## Module content

The content of the module will be split into four areas:

Kelvin-Helmholtz Instability.  Inviscid instability of the interface between two parallel flows of different density. This example introduces the notion of linear instability.

Inviscid instability of parallel flows. Derive the Rayleigh equation, examine the stability of piecewise linear flows, examine the stability criteria for smooth flows, introduce critical layer analysis.

Viscous instability. Derive the Orr-Sommerfeld equation, study the linear stability of plane Poiseuille flow, look at the asymptotic theory in the inviscid limit, and the connection to Rayleigh’s equation.

Boundary layer flows. Uniform flow over a flat plate, flow past a plate with arbitrary slip velocity, stability of boundary layers.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 mins) 20
Examination End-of-Semester Examination (2 hours) 80

NA

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of the methods required to solve complex hydrodynamic stability problems.

• Subject knowledge through the recall of definitions as well as explaining why particular flows are unstable, and concepts such as boundary layer receptivity, in physically relevant situations.

• Analytic ability through the solution of unseen and seen similar problems in the coursework assignment and exam.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 3.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 5.

Formative assessment
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback
Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal feedback given in lectures. Students also receive verbal feedback in office hours.

## Module aims

• Introduce students to the concept of fluid stability in parallel flows.
• Enable students to determine dispersion relations for simple channel flows and interpret the results in terms of stable and unstable flows.
• Develop the mathematics required to study critical layers in inviscid flows.
• Illustrate the role of fluid stability in the real world, such as in engineering applications.
• Discuss how the techniques studied in this course can be applied to problems in fields such as engineering.

## Learning outcomes

 Attributes Developed 001 Students will derive the dispersion relation for simple piecewise linear channel flows and solve for their stability properties. KCT 002 Students will interpret the results of the dispersion relation and classify whether a flow is linearly stable or unstable. KC 003 Students will derive the two stability criteria for smooth velocity profiles and use these to determine when a smooth velocity profile is unstable or when it could be unstable. KCT 004 Students will be able to derive and interpret the inviscid Tollmien critical layer solutions. KCT 005 Students will understand the significance of boundary layers in high speed fluid flows and will be able to formulate the basic velocity profiles within a boundary layer. They will also understand how these are equations are solved using numerical techniques. KCPT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A thorough account of hydrodynamic stability and its application to problems in areas such as engineering.

• Experience (through demonstration) of the methods and techniques used to solve problems in hydrodynamic stability.

The learning and teaching methods include:

• Three one-hour lectures per week for eleven weeks, in which notes can be taken. Lectures are delivered using blackboards/ whiteboards and/or visualizers for real-time presentation. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.

• There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MATM070

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM028 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: Students are equipped to address fluid dynamics complexities. This skill is useful in various fields including environmental engineering, aerospace, and energy.

Global and Cultural Capabilities: Student engagement in discussions during lectures naturally cultivates the sharing of the different cultures from which the students originate.

Resourcefulness and Resilience: The complexity of the boundary layer problems in MATM028 cultivates creative problem-solving, nurturing resourcefulness. Understanding hydrodynamic stability instills resilience, preparing students to navigate complex scenarios.

Sustainability: Students acquire expertise in boundary layer problems. This plays a valuable role in the design of environmentally sustainable infrastructure and in mitigating ecological stressors. MATM028 empowers students to tackle fluid dynamics challenges, promoting sustainable approaches in water resource management and environmental conservation.

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.