A Brief Analysis of Fluid Dynamics in Exhaust Systems

Billowing Hot Air: A Brief Analysis of Fluid Dynamics in Exhaust Systems
By: Tyler "BigTyla" Williams

Introduction

I remember when I first got into the car scene. I was eager to learn everything there was to know about enhancing the performance of my automobile. For most people the first thing that comes to mind when looking for that first modification is the exhaust system. It’s probably the most recognized modification even to those not familiar with the performance industry, especially with all those big trucks with obnoxious pipes and the token Honda Civic in with a coffee can on the back running around in every city.

If you were like me in your early “car guy” stages, you probably ran a Google search on the best way to improve your exhaust system. You read articles stating that bends in your exhaust were bad, and that if you had to bend your piping, the mandrel bend was the way to go. You also started seeing terms like backpressure, scavenging, and turbulence thrown around. If you got into header design, you also read certain setups gave more low-end torque while others gave high-end horsepower. A majority of people take the industry’s word for it and end their search there, but if you’re inquisitive like me, you’re left wondering why these statements are true (and even if some of the statements made are false). Unfortunately, there is very little explanation to be found on the internet for these statements, so many people are left scratching their head.

The other day I was brushing up on turbulence when I began to recall some classic pipe flow problems used in aerodynamics academia. One thing lead to another and suddenly I had the urge discuss the finer physics in pipe flow that are usually reserved for graduate-level courses in aerospace engineering. Since this topic requires high-level understanding to be fully appreciated (indeed, once could spend an entire semester lecturing on wall-bounded flows!), I will only be giving a cursory analysis on these concepts.

The Science Behind Your Exhaust System

What is the goal of an exhaust system?

The exhaust system is designed to deliver hot, harmful fumes from the engine bay and vent it to the atmosphere more openly. It should perform this function as efficiently as possible, i.e. it should result in the lowest loss in horsepower possible. Indeed, one can think of the exhaust system as one large restrictor. In terms of performance, it serves no purpose except to route the hot air away from the engine to make sure it ingests cool air. A consequence of the existence of an exhaust system is a reduction in the sound of the engine. This is due to the restrictions in the exhaust system causing a reduction in flow from the exhaust ports of the engine to the end of the exhaust pipe. In the automotive industry, this flow reduction is referred to as back pressure. In the interest of reducing noise further, mufflers are installed on the exhaust. Much of their noise-reducing qualities come from the implementation of a variety of acoustic techniques to quiet the sound of the combustion of the engine. Further reduction in sound comes as a result of acoustic wave damping by the piping.

An aside on back pressure

Officially, back pressure is the resistance of a fluid to obstructions in wall-bounded flows. This resistance causes a reduction in flow in the exhaust system. To more easily illustrate this, imagine pumping air through a constant diameter pipe with various obstructions. If you were to measure the dynamic pressure at the beginning and end of the pipe, the difference in the two measurements would be the back pressure. Hence, the ultimate goal in performance exhaust design is to have as little back pressure as possible.

There is a great deal of confusion within the performance community regarding back pressure, and this is the result of the term pressure being thrown around without clarification. When someone refers to a drop in pressure in an exhaust system, they are referring to dynamic pressure. This drop in dynamic pressure is the result of a reduction in exhaust velocity due to obstructions and friction of the pipe. It is imperative that you understand that Bernoulli’s principle does not apply in flow through long pipes, and therefore the dynamic pressure drop results in a drop in total pressure as well! For more information on the effect of friction on pressure loss in a pipe, see the Darcy-Weisbach equation

Exhaust flow is always turbulent

In reading some articles by enthusiasts regarding exhaust flow, I stumbled upon a few of them referring to laminar flow in the pipe. This was quite troubling to me since the flow in the pipe will never be laminar! To prove this, I introduce the Reynolds number, a ratio of inertial forces to viscous forces defined as Re = (V * L) / nu , where V is the fluid velocity, L is the characteristic length (diameter of the pipe for pipe flow), and nu is the kinematic viscosity of the fluid.

The Reynolds number is a well-known value in aerodynamics and has a variety of uses. One of these uses is defining transition of a flow from laminar to turbulent in a variety of flow fields. This is done empirically and can therefore vary based on the characteristics of the flow field, such as surface roughness, flow uniformity, and for wall-bounded flows, the shape of the wall. For pipe flow, transition to turbulence occurs generally at Re > 2300 with fully developed turbulence occurring at Re > 4000. Let us take a look at what kind of Reynolds numbers we will see in our car.

If we take the density of the gas to be 1.2 kg/m^3 (a fair guess assuming stoic air/fuel ratio), use a characteristic length of 0.075 meter (close to 3 inches) and take the dynamic viscosity to be 3 * 10^-5 kg/m-s (assuming 400 degrees Celcius), we get a Reynolds number of 27,000V. This means that for this flow to remain fully laminar, our exhaust velocity would have to be about 0.08 m/s. You can’t achieve that even at idle! Hence, your exhaust flow will be turbulent in all conditions (except when it’s off).

Why turbulence in exhaust is bad

You’ll often hear of products claiming to reduce turbulence in the exhaust resulting in a net gain in horsepower. But what is the turbulence doing that’s so bad? The main thing you need to know about turbulence is that it is inherently unstable and chaotic. One of the consequences of this trait is that skin friction for turbulent flows is much greater than in the laminar case. The result of this is in pipe flow is increased drag, which leads to greater back pressure. While bends and obstructions in the exhaust system yield lower exhaust velocity, they also result in higher turbulent intensity which exacerbates the losses.

Deeper analysis of bends in exhaust

We’ve already covered the fact that bends increase turbulence and decrease flow velocity. One thing that I have not seen covered by other enthusiasts is a physical explanation of what actually happens to the fluid going through a bend that causes these traits to occur.

Assume that we have a mandrel-bent pipe (a pipe with constant radius throughout the bend) with a relatively short bend so that viscous diffusion effects are negligible. As the gas approaches the bend, the fluid must be accelerated inward, so a streamwise total pressure gradient is established in the flow field. This is why bends increase back pressure and reduce exhaust velocity at the end of the pipe. Effects on turbulence come from the secondary flow resulting from the pressure gradient. Slower fluid is present on the inside of the pipe while faster fluid is kept outside. This introduces a rotation relative to the mean streamline of the flow, resulting in vorticity in the streamwise direction. This vorticity increases the instability of the flow.

Applying the Science

Now that we have a greater appreciation for the fluid dynamics occurring behind the scenes, let us apply that science practically. The following discussion will involve less fluid dynamics and will instead concentrate on exhaust design. Those of you more familiar with exhaust systems will recognize many of the concepts discussed here, so this will serve as a nice review for you.

Tuning pipe diameter based on power curve

Choosing the correct exhaust pipe diameter is pivotal when attempting to maximize performance. A pipe too small will result in significant back pressure due to high exhaust velocities (see the Darcy-Weisbach equation). A pipe too large will result in insufficient fluid velocity, causing inertial forces in the pipes to become less dominant over the viscous forces. Furthermore, gas in a larger pipe will cool more rapidly as it traverses downstream due to the increased surface area of the pipe (see thermal radiation), causing a decrease in downstream pressure. Relating this to performance (and as stated quite thoroughly in the performance community), a smaller exhaust will give great low-end power as it will move the exhaust optimally at lower engine speeds, but will also net poorer high-end power due to the significant back pressure at higher exhaust velocities. A larger exhaust will do the opposite. Ultimately you want to choose a diameter that gives you the most area under the power curve.

Minimizing heat loss

Perhaps the most overlooked factor in increasing engine performance is minimizing heat loss in the exhaust. Thermodynamics tells us that more heat equals more power, so we want to conserve as much heat as possible in our exhaust system! This is especially effective in turbocharged vehicles because the conserved heat in the exhaust manifold translates to conserved pressure. The more pressure we conserve in the system, the easier it is to drive that turbine! Exhaust heat can be conserved via thermal spraying, exhaust wrap, or increasing wall thickness of the exhaust pipe. As stated earlier, smaller piping can also reduce heat loss.

Tuning exhaust runner length and diameter

Header design is a complex area that requires discussion all its own. There are a couple more factors to consider in header design. In particular, flow reversion and exhaust pulse tuning become a concern. Flow reversion occurs in an exhaust runner when its associated piston travels downward and creates a vacuum in the cylinder. This pressure differential draws hot exhaust gases into the combustion chamber and contaminates the air/fuel mixture, causing a reduction in power. Reversion is often combated by using a pipe diameter equal to or slightly greater than the exhaust port on the cylinder head. Pulse tuning refers to designing the exhaust runners such that the exhaust gases from combustion of one cylinder immediately precede another. This is why engineers attempt to implement equal-length runners in their header designs. The trick is getting just the right length for a particular engine, something that requires a lot of time on the dynamometer and flow bench (most companies don’t have the experience or resources to do this). Indeed, the most efficient design could have different lengths for every runner depending on firing order! Adding a turbocharger to the equation complicates the problem even further as turbo lag becomes a concern. Optimally, a different header design would exist based on the turbocharger the user intends to install!

Conclusion

All of this information leads to conclusions that are well-documented, but allow me to reiterate for the sake of closure.

1) An ideal exhaust system has as few bends as possible.
2) Turbulence will exist in your exhaust system at all engine conditions.
3) We should work to reduce turbulence in exhaust as much as possible. To remedy, see #1.
4) Heat should be conserved as much as possible in an exhaust system.
5) Back pressure is bad. To remedy, see #1 and #4.
6) Exhaust pipe diameter should be selected based on the size that yields the most area under the power curve.
7) Tuning exhaust runners is extremely complex and expensive and involves many variables, and is also dependent on turbocharger size.

I hope you enjoyed this more thorough explanation of fluid dynamics and how it relates to exhaust systems. I anticipate it will serve as a basic guide for the budding automotive enthusiast wanting to delve deeper into the physics of this problem. I also hope this discussion serves as a catalyst for the design of more efficient exhaust systems on our platform!