# Abstract

We present a new explanation of the generation of lift during a wing stroke of flapping flight based on computing turbulent solutions of the Navier-Stokes equations at moderate to large Reynolds numbers. The new theory suggest a resolution of the enigma of why bumblebees can fly.

**First prompted by the fact of aviation, I have applied the laws of the resistance of air to insects, and I arrived at the conclusion that their flight is impossible.**(Antoine Magnan in Le Vol des Insectes)

Based on the Knols

Bumblebee, the Movie. Listen to The Flight of the Bumblebee.

## Existing Theory Insufficient

*What force does an insect wing generate” has been the driving force behind much of the research described here. An innocent question, but a difﬁcult one to answer. The difﬁculty is, in part, due to our lack of simple theories of unsteady ﬂuids stirred by a moving geometry. In search for the answer, we are forced to explore solutions beyond the classical theories and develop appropriate tools. One hopes that the insight gained in studying insect ﬂight might lead to our ﬁnding efﬁcient ways to interact with ﬂuids. Besides insects, birds, ﬁsh, leaves, ﬂags, kites, sails, oars, and heart valves, all live in ﬂuids and encounter a similar set of problems*[1].*Understanding force generation on a ﬂapping wing, though a difﬁcult feat, is only a beginning of our understanding of insects or ﬂapping ﬂight in nature as a whole. “Why do insects or birds ﬂap their wings the way they do? ” and “how does ﬂapping ﬂight come about in the course of evolution? ” For us who are bound to Earth, to ﬂy like birds may always be a temptation. Fundamental to such an endeavor are efﬁciency and stability. “Can ﬂapping ﬂight be more efﬁcient and stable than a ﬁxed-wing ﬂight?” [1].**A detailed aerodynamic analysis was used to show that quasi steady aerodynamic mechanisms are inadequate to explain even fast forward fligh*t.[2]

*Engineers say they can prove that a bumblebee can’t fly. And if you apply the theory of fixed wing aircraft to insects, you do calculate they can’t fly. You have to use something differen. If you treat a bird wing like an airplane wing and at any given time calculate the speed and lift, then sum it up over the entire stroke, it works fairly well to explain how the bird can stay aloft. With insect flight it fails miserably Dickinson.**Insects use three distinct but interacting techniques to gain lift: delayed stall, rotational circulation and wake capture*Dickinson.*We discovered that the flapping motion causes the leading edge vortex to spiral out to the wingtip, siphoning off the vortex and delaying stall. The augmented lift, coupled with the delayed stall, is the principle mechanism that insects use for generating lift…An inappropriate use of the quasi-steady assumption is certainly one way to ‘prove’ that bumblebees cannot fly.*

*Rapid oscillations pose one of the most difficult questions for fluid dynamics. Things become very messy*Z. Jane Wang.

## New Theory

**aoa**(see larger plots ) of a Naca0012 3d wing:

**aoa**= 20 with a lift/drag = 3.

The basic mechanisms generating lift and drag of a wing for **aoa** < 15 can be described as follows:

We see potential flow (upper left) with zero lift/drag modified by low-pressure counter-rotating rolls of streamwise vorticity from instability mechanism at separation (upper right), switching the pressure on rear wing (lower left) to give both lift and drag (H high, L low pressure). Computational simulation with sideway of velocity and pressure and topview of streamwise vorticitity at 14 degrees angle of attack is show lower right.