7. Vector Field Visualization Vector data set Represent direction - PDF document

7. Vector Field Visualization Vector data set • Represent direction and magnitude • Given by a n- tupel (f 1 ,...,f n ) with f k =f k (x 1 ,...,x n ), n ≥ 2 and 1 ≤ k ≤ n • Specific transformation properties • Typically n = k = 2 or n = k = 3 • Visualization, Summer Term 03 VIS, University of Stuttgart 1 7. Vector Field Visualization Main application of vector field visualization is flow visualization • Motion of fluids (gas, liquids) • Geometric boundary condititions • Velocity (flow) field v ( x ,t) • Pressure p • Temperature T • Vorticity ∇× v • Density ρ • Conservation of mass, energy, and momentum • Navier-Stokes equations • CFD (Computational Fluid Dynamics) • Visualization, Summer Term 03 VIS, University of Stuttgart 2

7. Vector Field Visualization Flow visualization based on CFD data Visualization, Summer Term 03 VIS, University of Stuttgart 3 7. Vector Field Visualization Flow visualization – classification • Dimension (2D or 3D) • Time-dependency: stationary (steady) vs. instationary (unsteady) • Grid type • Compressible vs. incompressible fluids • In most cases numerical methods required for flow visualization • Visualization, Summer Term 03 VIS, University of Stuttgart 4

7.1. Vector Calculus Review of basics of vector calculus • Deals with vector fields and various kinds of derivatives • Flat (Cartesian) manifolds only • Cartesian coordinates only • 3D only • Visualization, Summer Term 03 VIS, University of Stuttgart 5 7.1. Vector Calculus Scalar function ρ ( x ,t) • ( )  ρ    ∂ x , t ∂  ∂   ∂  x x ( ) ( ) ( ) ∇ ρ = ρ = ρ x , t ∂ x , t ∂ x , t Gradient     • ∂ ∂ y y     ( ) ∂ ρ ∂ x , t     ∂ ∂ z z Gradient points into direction of maximum change of ρ ( x ,t) • ( ) ( ) ∆ ρ = ∇ • ∇ ρ x , t x , t Laplace • ( ) ( ) ( ) = ∂ 2 ρ + ∂ 2 ρ + ∂ 2 ρ x , t x , t x , t ∂ 2 ∂ 2 ∂ 2 x y z Visualization, Summer Term 03 VIS, University of Stuttgart 6

7.1. Vector Calculus Vector function v ( x ,t) •   ∂ v ∂ v ∂ v   ∂ x x ∂ y x ∂ z x Jacobi matrix • ( ) = ∇ =   J v x , t ∂ v ∂ v ∂ v (“Gradient tensor”) ∂ x y ∂ y y ∂ z y   ∂ v ∂ v ∂ v   ∂ x z ∂ y z ∂ z z Divergence • ( ) ( ) ( ) ( ) ( ) = ∇ ⋅ = + + div v x , t v x , t ∂ v x , t ∂ v x , t ∂ v x , t ∂ x ∂ y ∂ z x y z Visualization, Summer Term 03 VIS, University of Stuttgart 7 7.1. Vector Calculus Gauss theorem (divergence theorem) • ∫ ∫ ∇ ⋅ = ⋅ v dV v d A V S v volume V surface S Visualization, Summer Term 03 VIS, University of Stuttgart 8

7.1. Vector Calculus y Properties of divergence: • div v is a scalar • div v ( x 0 ) > 0: v has a source in x 0 • div v ( x 0 ) < 0: v has a sink in x 0 • x div v ( x 0 ) = 0: v is source-free in x 0 • Describes flow into/out of a region • y x Visualization, Summer Term 03 VIS, University of Stuttgart 9 7.1. Vector Calculus Continuity equation • Flow of mass into a volume V with surface S • ∫ − ρ ⋅ v d A S Change of mass inside the volume • ∂ ∂ ρ ∫ ∫ ρ = dV dV ∂ ∂ t t V V Conservation of mass • ∂ ρ ∫ ∫ = − ρ ⋅ dV v d A ∂ t V S Visualization, Summer Term 03 VIS, University of Stuttgart 10

7.1. Vector Calculus Continuity equation (cont.) • Application of Gauss theorem • ∂ ρ ∂ ρ   ( ) ∫ ∫ ∫ + ρ ⋅ = + ∇ ⋅ ρ = dV v d A  v  dV 0 ∂ ∂ t  t  V S V Above equation must be met for any volume element • Yields • ∂ ρ ( ) + ∇ ⋅ ρ = v 0 ∂ continuity equation t in differential form = ρ j v current Visualization, Summer Term 03 VIS, University of Stuttgart 11 7.1. Vector Calculus Curl • ( ) ( )   − ∂ v x , t ∂ v x , t   ∂ ∂ y z z y ( ) ( ) ( ) ( ) = ∇ × =  −  curl v x , t v x , t ∂ v x , t ∂ v x , t ∂ z x ∂ x z  )  ( ) ( − ∂ ∂ v x , t v x , t   ∂ y ∂ x x y Stokes theorem • ∫ ∫ ∇ × ⋅ = ⋅ v d A v d s surface S v S C closed curve C Visualization, Summer Term 03 VIS, University of Stuttgart 12

7.1. Vector Calculus Properties of curl: • Describes vortex characteristics in the • y flow From non-curl free flow • ∇ × ≠ v 0 x can follow that ∫ ∇ × ⋅ ≠ v d A 0 S and ∫ ⋅ ≠ v d s 0 C ∇ × v Vorticity • Visualization, Summer Term 03 VIS, University of Stuttgart 13 7.1. Vector Calculus Decomposition of a vector field (Helmholtz theorem) • Divergence-free (transversal) part • Curl-free (longitudinal) part • Decomposition: • = + v ( x ) v ( x ) v ( x ) c d with ∇ × = v c ( x ) 0 ∇ • = v d ( x ) 0 and Other way round: v is uniquely determined by divergence-free and curl- • free parts (if v vanishes at boundary / infinity) Visualization, Summer Term 03 VIS, University of Stuttgart 14

7.1. Vector Calculus Decomposition of a vector field (cont.) • Suppose we have a representation of v by the Poisson equation • = − ∆ v ( x ) z ( x ) Curl-free part written as a gradient: • = −∇ v ( x ) u ( x ) c = ∇ • u ( x ) z ( x ) with (scalar) potential u Divergence-free part written as a curl • = ∇ × v ( x ) w ( x ) d = ∇ × w ( x ) z ( x ) vector potential w with Visualization, Summer Term 03 VIS, University of Stuttgart 15 7.2. Characteristic Lines Types of characteristic lines in a vector field: • Stream lines: tangential to the vector field • Path lines: trajectories of massless particles in the flow • Streak lines: trace of dye that is released into the flow at a fixed position • Time lines (time surfaces): propagation of a line (surface) of massless • elements in time Visualization, Summer Term 03 VIS, University of Stuttgart 16

7.2. Characteristic Lines Stream lines • Tangential to the vector field • Vector field at an arbitrary, yet fixed time t • Stream line is a solution to the initial value problem of an ordinary • differential equation: ( ) d L u ( ) ( ( ) ) = = L 0 x , v L u , t 0 d u initial value ordinary differential equation (seed point x 0 ) Stream line is curve L ( u ) with the parameter u • Visualization, Summer Term 03 VIS, University of Stuttgart 17 7.2. Characteristic Lines Path lines • Trajectories of massless particles in the flow • Vector field can be time-dependent (unsteady) • Path line is a solution to the initial value problem of an ordinary differential • equation: ( ) d L u ( ) ( ( ) ) = = L 0 x , v L u , u 0 d u Visualization, Summer Term 03 VIS, University of Stuttgart 18

7.2. Characteristic Lines Streak lines • Trace of dye that is released into the flow at a fixed position • Connect all particles that passed through a certain position • Time lines (time surfaces) • Propagation of a line (surface) of massless elements in time • Idea: “consists” of many point-like particles that are traced • Connect particles that were released simultaneously • Visualization, Summer Term 03 VIS, University of Stuttgart 19 7.2. Characteristic Lines Comparison of path lines, streak lines, and stream lines • t 0 t 1 t 2 t 3 stream line for t 3 path line streak line Path lines, streak lines, and stream lines are identical for steady flows • Visualization, Summer Term 03 VIS, University of Stuttgart 20

7.2. Characteristic Lines Difference between Eulerian and Lagrangian point of view • Lagrangian: • Individual particles • Can be identified • Attached are position, velocity, and other properties • Explicit position • Standard approach for particle tracing • Eulerian: • No individual particles • Properties given on a grid • Position is implicit • Visualization, Summer Term 03 VIS, University of Stuttgart 21 7.3. Arrows and Glyphs Visualize local features of the vector field: • Vector itself • Vorticity • Extern data: temperature, pressure, etc. • Important elements of a vector: • Direction • Magnitude • Not: components of a vector • Approaches: • Arrow plots • Glyphs • Visualization, Summer Term 03 VIS, University of Stuttgart 22

7.3. Arrows and Glyphs Arrows visualize • Direction of vector field • Orientation • Magnitude: • Length of arrows • Color coding • Visualization, Summer Term 03 VIS, University of Stuttgart 23 7.3. Arrows and Glyphs Arrows • Visualization, Summer Term 03 VIS, University of Stuttgart 24