LAGRANGIAN & EULERIAN DESCRIPTIONS Lagrangian Approach: Akışkanın hareketini Lagrange bakış açısı ile belirlemek demek, her bir akışkan parçasının yaptığı hareketi teker teker belirlemek demektir Describe the fluid particle’s motion with time. The path of a particle: r(t) = x(t) i + y(t) j + z(t) k i, j, k: unit vectors Velocity of a particle : V(t) = dr(t) / dt = u i + v j + w k Eulerian Approach: ■ Akım alanının her noktasında, hareket ile ilgili büyüklüklerin (hız, basınç, vb.) zamanla nasıl değiştiklerini belirleyelim Imagine an array of windows in the flow field: Have information for the fluid particles passing each window for all time. In this case, the velocity is function of the window position (x, y, z) and time. u = f 1 (x, y, z, t) v = f 2 (x, y, z, t) w = f 3 (x, y, z, t) Eulerian approach is generally favored
Streamlines & Flow Patterns Flow Pattern: Construction of streamlines showing the flow direction Streamlines (light blue): Local velocity vector is tangent to the streamline at every point along the line at a single instant. Flow through an opening in a tank & over an airfoil section.
Streamline & Pathline Streamline: line drawn through flow field such that local velocity vector is tangent at every point at that instant – Tells direction of velocity vector – Does not directly indicate magnitude of velocity Pathline: shows the movement of a particle over time ► In unsteady flow, all can be distinct lines. ► The latter two tells us the history of flow as the former indicates the current flow pattern.
3.3. AKIM ÇiZGiSi VE AKIM BORUSU ■ Hız vektörlerine teğet olarak çizilen eğrilere akım çizgileri denir. Akım çizgisi ile Şekil 3.1 de tanımlanan yörünge aynı şey midir? Akım çizgisi ve yörünge ancak, zamanla-değişmeyen akım halinde üst üste düşerler. Zamanla-değişen akım halinde bunlar farklı farklı şeylerdir.
Examples... Predicted streamline pattern over the Volvo ECC prototype. Pathlines of floating particles.
TYPES OF FLOW Uniform: Velocity is constant along a streamline (Streamlines are straight and parallel) Non-uniform: Velocity changes along a streamline (Streamlines are curved and/or not parallel) Express velocity V = V(s,t) Vortex flow
Steady: streamline patterns are not changing over time zamanla-değişmeyen akım (permanan akım) Unsteady: velocity at a point on a streamline changes over time Flow patterns can tell you whether flow is uniform or non-uniform, but not steady vs. unsteady… Why? Because streamlines are only instantaneous representation of the flow velocity. TYPES OF FLOW
LAMINAR & TURBULENT FLOW (a)Experiment to illustrate the type of the flow (b) Typical dye streaks for different cases (a) (b)
DIMENSIONALITY OF FLOW FLIED → Characterized by the number of spatial dimensions needed to describe velocity field. 1-D flow: Axisymmetric uniform flow in a circular duct 2-D flow: Uniform flow in a square duct 3-D flow: Uniform flow in an expanding square duct
FLOW ACCELERATION (rate of change of velocity with time ) Consider a fluid particle moving along a pathline... There are two components of acceleration: Tangential to pathline a t : the time-dependent acceleration related to change in speed. Normal to pathline a n : the centripetal acceleration related to motion along a curved pathline.
Flow Acceleration Local acceleration – occurs when flow is unsteady (direction or magnitude is changing with respect to time) Convective acceleration – occurs when flow is nonuniform (acceleration can depend on position in a flow field) Local acceleration – occurs when flow is unsteady Centripetal acceleration – occurs when the pathline is curved (normal to the pathline & directed toward the center of rotation)
Example: Convective Acceleration The nozzle shown below is 0.5 meters long. Find the convective acceleration at x = 0.25 m. The equation describing velocity variation is provided below.