# AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

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1 AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B-1: Mathematics for Aerodynamics B-: Flow Field Representations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis AE301 Aerodynamics I : List of Subjects Concept of Streamline Strain and Angular Velocity Angular Velocity and Vorticity Rotational v.s. Irrotational Flows Circulation of a Flow Field Kutta-Joukowski Theorem Stream Function Velocity Potential Inviscid and Incompressible Flow Equations Laplace s Equation

2 Page 1 of 15 Concept of Streamline CONCEPT OF STREAMLINE A streamline is a line whose tangent at any point is in the direction of the velocity vector at that point. Mathematically, this means: ds V 0. Let us examine this equation in 3-D Cartesian coordinate system: ds dxˆi dyˆj dzk ˆ and V uˆi vˆj wk ˆ ˆi ˆj kˆ Therefore, ds V dx dy dz ˆi wdy vdz ˆj udz wdx k ˆ vdx udy u v w EQUATION OF STREAMLINE In order to satisfy the condition of streamline: ds V 0, its each component must all equal to zero. Therefore: wdy vdz 0 (y-z plane) => dz dy w v udz wdx 0 (x-z plane) => dz w dx u vdx udy 0 (x-y plane) => dy dx v u

3 Page of 15 Strain and Angular Velocity THE MOTION OF A FLUID ELEMENT The motion of a fluid element along a streamline is a combination of translation and rotation. In addition, the shape of the element can become distorted ( angular deformation ) as it moves. The presence of viscosity causes both rotation and angular deformation, and the flow field is defined as rotational flow. If the viscosity within the flow field is ignored ( inviscid flow ), the motion of a fluid element can be simplified only for translation : this is called, irrotational flow. ANGULAR VELOCITY OF A FLUID ELEMENT (IN -D CARTESIAN: X-Y PLANE) When a fluid element rotates (and also being distorted due to its rotation), the magnitude of rotation of an element (in x-y plane) can be defined by the angular velocity as: 1 1 z t t Note that the angular velocity (rotation) can be related to the rate of shear strain (deformation) as: (traveling distance, due to velocity difference) = (arc length, due to rotation) The edge dy : u dy t dy( ) 1 u 1 => y t y The edge dx : v dx t dx( ) v => x t x 1 v u Therefore, the angular velocity of a fluid element (in x-y plane) is: z x y

4 Page 3 of 15 Angular Velocity and Vorticity ANGULAR VELOCITY VECTOR OF A FLUID ELEMENT (3-D CARTESIAN) Repeating a similar procedure in y-z and x-z planes, one can obtain: 1 w v x y z (y-z plane) and 1 u w y (x-z plane) z x The angular velocity is a vector (thus, 3-components in 3-D Cartesian coordinate system), such that: ˆ ˆ ˆ 1 w v ˆ u w ˆ v u ˆ ω xi yjzk i j k (angular velocity vector) y z z x x y Let s recall: the curl of the velocity field is... V VORTICITY curl? In aerodynamics, the curl of the velocity field is equal to the twice of the angular velocity. w v u w v u ξ ω i j k V V y z z x x y Mathematically: ˆ ˆ ˆ curl This is defined as: vorticity of a given flow field, in aerodynamics. ξ curlv V (vorticity of flow field)

5 Page 4 of 15 Rotational v.s. Irrotational Flows Irrotational Flow: V 0 ROTATIONAL V.S. IRROTATIONAL FLOW We can categorize flow field of aerodynamics into two distinctively different types, based on the vorticity in the flow field: IDEAL (OR POTENTIAL ) FLOW FIELD If the flow field is irrotational, the governing equation of the flow field can be significantly simplified. As a result, one can model and analyze basic behaviors of the flow field by simple mathematical formulations. If the flow field is: (i) steady, (ii) no body force, (iii) incompressible, (iv) inviscid, (v) and irrotational (because inviscid ): the flow field is called, ideal or potential flow field. Ideal (or potential) flow fields can be analyzed mathematically (100% pure theory). This is commonly referred to be: theoretical aerodynamics by potential flow field analysis.

6 Page 5 of 15 Circulation of a Flow Field + C V ds V ds S CIRCULATION OF A FLOW FIELD Flow field may have a circulation. The circulation is a very fundamental tool, in theoretical aerodynamics, to calculate aerodynamic lift within an ideal (or potential) flow field. The presence of a body (such as an airfoil ) within a flow field will cause circulation within the flow field. If we calculate the amount of circulation within the flow field, we can calculate the amount of lift of the body. Mathematically, the circulation can be calculated by either a line integral over a given line path (C) defined by the presence of the body, or an area integral over the area (S) bounded by the line path. C Vds V ds (circulation) S CIRCULATION THEORY: CLASSICAL THEORY OF AERODYNAMICS The circulation theory is a classical mathematical formulation for the calculation and the aerodynamic lift of a body within a given flow field. This is very well-established mathematical model based flow field analysis of an ideal (or potential) flow field (pure theory basis). In circulation theory, an aerodynamic lift is generated due to the time rate of change of airflow in the downward direction, due to the amount of circulation in the flow field, generated by the presence of the body. Kutta-Joukowski Theorem: (probably) the most important theorem in theoretical aerodynamics

7 Page 6 of 15 Kutta-Joukowski Theorem V L L' V C KUTTA-JOUKOWSKI THEOREM If a circulation of the -D flow field is known, the lift per unit depth ( L ') can be calculated from freestream velocity and density of the flow field: L' V This is the famous Kutta-Joukowski theorem for an ideal (or potential) flow field. According to this theorem, you can calculate the lift of a body, if you know the circulation of the flow field around the body (which is, generated due to the presence of the body itself in the flow field).

8 Page 7 of 15 Class Example Problem B--1 Related Subjects... Circulation Consider the -D velocity field: V y x ( x y ) i ( x y ) j If the velocity is given in units of ft/s, calculate the circulation around a circular path of radius r. centered around the origin. + r C Circulation can be calculated as: Vds VdS C S However, for irrotational flows, the vorticity is zero: V 0 => VdS 0dS =??? S S y The flow field is given as: ˆ x V i ˆj x y x y y x where, u and v x y x y Let us convert into the polar coordinate system. Note that: x rcos, y rsin, and x y r Vr ucos vsin, and V u sin vcos y rsin sin x rcos cos u and v x y r r x y r r sin cos sin cos 1 V r cos sin 0 and V sin cos r r r r r 1 The circulation is: d V ˆ rd ˆ V s rd d C e e r (ft /s) Note that the circulation is independent to the radius (constant throughout the flow field, in this case). This is called the point vortex flow (one of the elementary potential flow models)

9 Page 8 of 15 Stream Function (1) STREAM FUNCTION Staring from the equation of streamlines (in x-y plane), let s derive the stream function. dy v => u x, ydy v x, ydx => u x, ydy vx, ydx dx u f x, y c => Let us denote this function f, x y by the symbol, called the stream function. A unique equation for a streamline is given by setting the stream function equal to a unique constant., c, c3,... represent the streamlines of the flow field. c 1 STREAMLINES OF A FLOW FIELD If u and v are known functions of x and y, then the equation of streamlines can be integrated to yield: f x, y c where, c is an arbitrary constant (will be different value for each streamline). f x, y c is called, the equation of streamlines. Let s properly understand that this generates multiple equations (a uniquely different equation for each streamline, for a different value of c). f x, y is called, the stream function (a function, that can generate multiple equations of streamlines with multiple constants c assigned to this function).

10 Page 9 of 15 Stream Function () 1. = constant gives the equation of a streamline.. The stream function can be defined in -D flows only. 3. If the stream function is known, the flow field can be described by: u 1 v V y r V x r r CONCEPT OF STREAM FUNCTION Stream function is a mathematical function that can represent a -D steady flow field. For incompressible flow field: can be defined as the stream function. The dimension of stream function is: kg/(sm): mass flow rate per unit depth. m 3 /(sm): volume flow rate per unit depth. The difference in stream function represents: : mass flow (per unit depth) between two adjacent streamlines. : volume flow (per unit depth) between two adjacent streamlines.

11 Page 10 of 15 Velocity Potential (1) VELOCITY POTENTIAL (1) Recall that an irrotational flow field is defined as a flow where the vorticity is zero at every point of ξ curl V V 0 the flow field, such that: Consider a scalar function, called the velocity potential. The vector identity can be written as: 0 This equation states that for an irrotational flow, there exists a scalar function such that the velocity is given by the gradient of. EQUIPOTENTIAL LINES OF A FLOW FIELD Similar (in concept) to streamlines, the equation of equipotential lines can be given as: f x, y c where, c is an arbitrary constant (will be different value for each line). f x, y c is called, the equation of equipotential lines. This generates multiple equations (a uniquely different equation for each line, for a different value of c). f x, y is called, the velocity potential (a function, that can generate multiple equations of equipotential lines with multiple constants c assigned to this function). VELOCITY POTENTIAL () The velocity potential is a function of a spatial coordinate: ( x, y, z), or ( r,, z) For a 3-D Cartesian coordinate system: uˆ vˆ w ˆ ˆ ˆ V i j k i j k ˆ x y z Therefore: u, v x z, and w y

12 Page 11 of 15 Velocity Potential () 1. = constant gives the equation of a equipotential lines.. The velocity potential can be defined for irrotational flow only. 3. The velocity potential can be defined for both -D and 3-D flows. 4. If the velocity potential is known, the flow field can be described by: CONCEPT OF VELOCITY POTENTIAL Velocity potential is a mathematical function that can represent a steady irrotational flow field. If a velocity potential can be defined for a flow field, there will be a significant simplification: Velocity components ( u, v, w ): 3 unknowns => velocity potential ( ): 1 unknown A flow field that can be described by a velocity potential is called, the potential flows. STREAMLINES AND EQUIPOTENTIAL LINES Lines of constant : streamlines Lines of constant : equipotential lines The slope of a =constant line is the negative reciprocal of the slope of a =constant line: means that streamlines and equipotential line are mutually perpendicular. IDEAL (OR POTENTIAL) FLOW FIELD ANALYSIS The flow field that can define velocity potential (steady and irrotational flow field) is called the potential flow field. In aerodynamics I (AE301), we define ideal flow field being incompressible potential flow field; meaning: (i) stready, (ii) no body force, (iii) inviscid, (iv) irrotational (because inviscid), (v) and incompressible. Ideal flow field is governed by the equation, called the Laplace s equation. Theoretical aerodynamics is a basic mathematical analysis, attempting to solve this Laplace s equation for an ideal flow field.

13 Page 1 of 15 Inviscid and Incompressible Flow Equations Bernoulli s Equation: Potential Flow: Pressure Coefficient: EQUATION SIMPLIFICATION PROCESS IN AERODYNAMICS The x-momentum equation: the governing equation Du p fx fx Dt x viscous Steady, inviscid flow with no body forces: Euler s equation Incompressible: Bernoulli s equation Irrotational: ideal (or potential) flows VELOCITY-PRESSURE COUPLING FOR AN IDEAL FLOW FIELD For an ideal flow field, the pressure coefficient can also be simplified as: 1 ( ) V V p p V V V C p 1 q 1 V V V If the velocity field is given, one can easily calculate the distribution of pressure (pressure field) within the flow field. This is called, the velocity-pressure coupling.

14 Page 13 of 15 Laplace s Equation (1) LAPLACE S EQUATION (1) Recall, from the conservation of mass (continuity): 0 V (differential form) t For steady 0 and incompressible ( constant t ) flow, this equation can be simplified to: V 0 (note: this is the continuity equation of steady & incompressible flow) (Mathematically, this is the Divergence of a velocity field ) LAPLACE S EQUATION () Starting from the continuity equation for steady and incompressible flow field: V 0 If the flow field is irrotational, a velocity potential can be defined, such that: V Combining these two equations, yields: 0 or, 0 This is the governing equation of potential flow field, called the Laplace s equation. THE LAPLACIAN OPERATOR Often, the Laplace s equation can be expressed by the special operator, called the Laplacian ( ): 0 or 0 (Laplacian of a scalar function is equal to zero)

15 Page 14 of 15 Laplace s Equation () POTENTIAL FLOW ANALYSIS Potential flow is a flow field that can define a velocity potential. Ideal flow field (in aerodynamics I) can be defined as a flow field of steady, inviscid, no body forces, incompressible, and irrotational. The ideal flow is governed by the Laplace s equation. The ideal flow field can be analyzed by pure theory (100% mathematical solution), which is essentially solving the Laplace s equation for a given flow field. LAPLACE S EQUATION In terms of velocity potential: 3-D Cartesian coordinate system, x, y, z x y z 0 : In terms of velocity potential: 3-D cylindrical coordinate system, r,, z 1 1 r r r r r z 0 : In terms of stream function: -D Cartesian coordinate system, xy, x y 0 In -D polar coordinate system, r, 1 1 r r r r r : 0 :

16 Page 15 of 15 Class Example Problem B-- Related Subjects... Laplace s Equation Examine Laplace s equation. What type of equation this is? What are the distinctive feature(s) of this type of equation? Laplace s equation is: (1) Partial differential equation (PDE) () with nd order (3) and linear The Laplace s equation is a linear nd order elliptic PDE. The distinctive features include: (A) Analytical solutions exist, with a given boundary conditions (called, the elementary potential flow solutions ). (B) These elementary potential flow solutions can be superimposed to construct another valid solution of the flow field. Laplace s equation is the governing equation for an ideal flow field. Therefore, one can analyze flow field behavior (i.e., velocity, pressure, aerodynamic forces & moments) of an ideal flow field by mathematically solve the Laplace s equation. This is the very foundation of aerodynamics (theoretical aerodynamics). In AE301, we ll start with very simple ideal flow field analysis (potential flow theory): flow over a circular cylinder without and with spin (circulation). Hopefully, later, you ll start using the aerodynamic analysis software (XFOIL / XFLR5). Note that the software is based entirely on the potential flow theory. Understanding the basic mechanism of aerodynamic analysis (the potential flow theory) is essentially understanding the theory of aerodynamics. Using the software AS A BLACK BOX is NOT an engineering practice! Anyone can do such a thing... the purpose of AE301 is to understand how the software works, and not using the software as a black box.

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