INTRODUCTION TO AERODYNAMICS
Dr. Sunil Manohar Dash
Assistant Professor
Department of Aerospace Engineering
Email: smdash@aero.iitkgp.ac.in
Introduction to Aerodynamics (AE20001/AE21201) 1
Kinematics of Fluid Flow
(AE20001/AE21201)
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community
Ask the community for help and clear up your study doubts
University Rankings
Discover the best universities in your country according to Docsity users
Free resources
Our save-the-student-ebooks!
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
3rd Semester notes of Intro to Aerodynamics, Schemes and Mind Maps of Aerodynamics
An introduction to aerodynamics, specifically focusing on the kinematics of fluid flow and the classifications of fluid flow. It also covers the Lagrangian and Eulerian methods of describing fluid motion, material derivative and acceleration, and flow field representation. mathematical equations and examples to illustrate the concepts.
Typology: Schemes and Mind Maps
2021/2022
1 / 62
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download 3rd Semester notes of Intro to Aerodynamics and more Schemes and Mind Maps Aerodynamics in PDF only on Docsity!
INTRODUCTION TO AERODYNAMICS
Dr. Sunil Manohar Dash
Assistant Professor
Department of Aerospace Engineering
Email: smdash@aero.iitkgp.ac.in
Introduction to Aerodynamics (AE20001/AE21201) 1
Kinematics of Fluid Flow
(AE20001/AE21201)
Kinematics of Fluid Flow
2 Fluid kinematics is the study on fluid motion in space and time without considering the force which causes the fluid motion. Solid in motion^ Fluid in motion Introduction to Aerodynamics (AE20001/AE21201)
Description of Fluid Motion
** Images are taken from various web sources 4 A. Lagrangian Method Using Lagrangian method, the fluid motion is described by tracing the kinematic behaviour of each fluid particle constituting the flow. Identities of the particles are made by specifying their initial position (spatial location) at a given time. The position of a particle at any other instant of time then becomes a function of its identity and time. is the position vector of a particle (with respect to a fixed point of reference) at a time t. is its initial position at a given time t = t 0 Analytical expression of the last statement : Above equation can be written into scalar components with respect to a rectangular cartesian frame of coordinates as: x = x(x 0 ,y 0 ,z 0 ,t) y = y(x 0 ,y 0 ,z 0 ,t) z = z(x 0 ,y 0 ,z 0 ,t) (where, x 0 ,y 0 ,z 0 are the initial coordinates and x, y, z are the coordinates at a time t of the particle) Hence in can be expressed as Lagrangian Introduction to Aerodynamics (AE20001/AE21201)
Description of Fluid Motion
** Images are taken from various web sources 5 A. Lagrangian Method The velocity and acceleration of the fluid particle can be obtained from the material derivatives of the position of the particle with respect to time. Therefore, In terms of scalar components, where u, v, w are the components of velocity in x, y, z directions respectively. Introduction to Aerodynamics (AE20001/AE21201)
Description of Fluid Motion
** Images are taken from various web sources 7 B. Eulerian Method The method was developed by Leonhard Euler. This method is of greater advantage since it avoids the determination of the movement of each individual fluid particle in all details. It seeks the velocity and its variation with time t at each and every location ( ) in the flow field. In Eulerian view, all hydrodynamic parameters are functions of location and time. Mathematical representation of the flow field in Eulerian method: where (^) and u = u ( x, y, z, t ) v = v ( x, y, z, t ) w = w ( x, y, z, t ) Therefore, Eulerian Introduction to Aerodynamics (AE20001/AE21201)
Description of Fluid Motion
8 The Eulerian description can be written as : C. Relation between Eulerian and Lagrangian Method The integration of above equation yields the constants of integration which are to be found from the initial coordinates of the fluid particles. Hence, the solution of above equation gives the equations of Lagrange as, Above relation are same as Lagrangian formulation. In principle, the Lagrangian method of description can always be derived from the Eulerian method. Introduction to Aerodynamics (AE20001/AE21201)
Classifications of Fluid Flow
10 Hydrodynamic parameters like pressure and density along with flow velocity may vary from one point to another and also from one instant to another at a fixed point. According to type of variations, categorizing the flow: Steady Flow and Unsteady Flow: Steady Flow A steady flow is defined as a flow in which the various hydrodynamic parameters and fluid properties at any point do not change with time. In Eulerian approach, a steady flow is described as, and Implications: Velocity and acceleration are functions of space coordinates only. In a steady flow, the hydrodynamic parameters may vary with location, but the spatial distribution of these parameters remain invariant with time. In the Lagrangian approach, time is inherent in describing the trajectory of any particle. In steady flow, the velocities of all particles passing through any fixed point at different times will be same. Introduction to Aerodynamics (AE20001/AE21201)
Classifications of Fluid Flow
11 Steady Flow Describing velocity as a function of time for a given particle will show the velocities at different points through which the particle has passed providing the information of velocity as a function of spatial location as described by Eulerian method. Therefore, the Eulerian and Lagrangian approaches of describing fluid motion become identical under this situation. Unsteady Flow An unsteady Flow is defined as a flow in which the hydrodynamic parameters and fluid properties changes with time. Introduction to Aerodynamics (AE20001/AE21201)
Classifications of Fluid Flow
13 Uniform and Non-uniform Flow Non-uniform Flow When the velocity and other hydrodynamic parameters changes from one point to another the flow is defined as non-uniform. Important points:
- For a non-uniform flow, the changes with position may be found either in the direction of flow or in directions perpendicular to it.
- Non-uniformity in a direction perpendicular to the flow is always encountered near solid boundaries past which the fluid flows. Reason: All fluids possess viscosity which reduces the relative velocity (of the fluid w.r.t. to the wall) to zero at a solid boundary. This is known as no-slip condition. Introduction to Aerodynamics (AE20001/AE21201)
Classifications of Fluid Flow
14 Four possible combinations of fluid flow
**1. Steady Uniform flow:
- Steady non-uniform flow:
- Unsteady Uniform flow:
- Unsteady non-uniform flow:** Flow at constant rate through a duct of uniform cross-section Flow at constant rate through a duct of non-uniform cross-section Flow at varying rates through a long straight pipe of uniform cross-section Flow at varying rates through a duct of non-uniform cross-section. Introduction to Aerodynamics (AE20001/AE21201)
Material Derivative and Acceleration
16 Expression of velocity components in the Taylor's series form: The increment in space coordinates can be written as - Substituting the values of in above equations, we have Introduction to Aerodynamics (AE20001/AE21201)
Material Derivative and Acceleration
17 In the limit , the equation becomes The above equations tell that the operator for total differential with respect to time, D/Dt in a convective field is related to the partial differential as: Introduction to Aerodynamics (AE20001/AE21201)
Material Derivative and Acceleration
19 Important points: In a steady flow, the temporal acceleration is zero, since the velocity at any point is invariant with time. In a uniform flow, on the other hand, the convective acceleration is zero, since the velocity components are not the functions of space coordinates. In a steady and uniform flow, both the temporal and convective acceleration vanish and hence there exists no material acceleration. Type of Flow Temporal Acceleration Convective Acceleration Steady Uniform flow 0 0 Steady non-uniform flow 0 exists Unsteady Uniform flow exists 0 Unsteady non-uniform flow exists exists Introduction to Aerodynamics (AE20001/AE21201)
Material Derivative and Acceleration
20 Example: Given the velocity field. Find the acceleration of fluid particle as a function of x,y,z and t, at (1,1,1) and time t= Introduction to Aerodynamics (AE20001/AE21201)