Graduate · Fourier incompressibility projection
Fourier incompressibility projection: A steep rational direction
Fourier incompressibility projection: investigate a steep rational direction with frequency component k₁ = 1; frequency component k₂ = 2.5.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
Watch the relationship
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Understand what you are seeing
The idea behind the motion.
At a nonzero Fourier frequency, incompressibility requires the velocity coefficient to be perpendicular to that frequency. The orthogonal projection removes only the longitudinal component. Rotating an input coefficient shows which information is kept and which is discarded without pretending to solve the full fluid dynamics. This investigation starts with Frequency component k₁ = 1; Frequency component k₂ = 2.5. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.
A relationship to keep
Pₖv=v−k(k·v)/|k|²; k·Pₖv=0
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Set up the mathematical model
Specify the nonzero frequency vector k=(a,b) and identify its perpendicular line. The starting case is “A steep rational direction.”
- STEP 2
Follow the changing quantity
Rotate a unit input vector, then subtract its component parallel to k. Follow the projected output and the removed component.
- STEP 3
Explain and test the result
Check k dot Pₖv equals zero and that projection never increases Euclidean norm. Applying the same projection twice leaves the result unchanged.
Your turn to explain
Make a prediction. Test your reasoning.
Keep Frequency component k₁ = 1; Frequency component k₂ = 2.5. Pause the timeline at 20%. Given projected x = -0.062, calculate projected y, k dot projected vector, projected norm. Show the substitution into the displayed formula.
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
For k=(1,2.5), |k|²=7.25. Subtract k(k·v)/|k|² from v=(0.309017,0.951057) to get (-0.061557,0.024623); the check (1)(-0.061557)+(2.5)(0.024623)=0 verifies perpendicularity. Results: Projected y: 0.025; k dot projected vector: 0; Projected norm: 0.066. Decimal values are rounded; retain the original parameters when checking.
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