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Helicopter Atmosphere Modeling

Helicopter performance varies significantly with atmospheric conditions. For example, aircraft drag increases with air density, rotor efficiency increases with density, and high temperatures decrease engine performance. For these reasons, evaluation of helicopter behavior must be done with specified atmospheric values. The intent of this article is to discuss these air properties.

Troposphere

Helicopters fly in the lowest layer of the atmosphere, called the troposphere. According to the International Standard Atmosphere (ISA), the troposphere extends to 11,000 m (36,090 ft) above mean sea level (MSL). Helicopter generally fly well below 20,000 ft, although some world record attempts have reportedly been flown over 40,000 ft.

Temperature, pressure, and density

There are three values we are interested in: temperature, pressure, and density denoted $t$, $p$, and $\rho$. Humidity plays only a small role and will be neglected (this is the norm). The three values are related by the equation below, where ($R$ is the gas constant).

$$\begin{equation} p = R \rho t \label{eq:thermoState} \end{equation}$$

Given this relationship, we only need to find two of the three values—the third value can always be computed from the equation above. Henceforth, we’ll focus on just temperature and pressure. This is the norm, because these are the easiest two values to measure in practice. For example, in flight test, temperature and pressure will be measured and density will typically be computed.

Change with altitude

An important consideration is how temperature and pressure change with altitude. For example, a pilot planning to fly a helicopter to a higher landing point needs to be confident the helicopter is capable of hovering or landing there. An engineer establishing operating limits needs to know how the atmosphere varies within the target flight envelope.

Temperature

Temperature fluctuates significantly throughout the atmosphere. No one predicts it perfectly in advance, and all we present here is a simple, empirical model for how temperature decreases with altitude. On any given day this may not be very accurate, but the point is that it’s roughly how temperature decreases with altitude on average within the troposphere, and is used in the ISA.

The model estimates that temperature decreases linearly with altitude in the troposphere. If $t_0$ is the temperature at MSL then the temperature $t$ at altitude $h$ above MSL is estimated by the equation below. ($\lambda = .001981$ C per ft is used in ISA.) $$ \begin{equation}t = t_0 - \lambda h \label{eq:tempLapse} \end{equation}$$

ISA specifies $t_0 = 15C$ on a “standard day,” but pilots and engineers often need to change $t_0$ to for hot or cold days. Rarely, they may also want to change $\lambda$ slightly. Equation \eqref{eq:tempLapse}, as simple as it is, is used almost universally.

Pressure

Most of what we said about temperature applies to pressure: it varies significantly, unpredictably, and we want a simple model for how it changes with altitude in the troposphere. Pressure also decreases with altitude, but using a linear rate (as done with temperature) introduces too much error. The slightly more complicated equation is provided below, giving the pressure $p$ at an altitude $h$ above MSL. As seen, it depends on the pressure $p_0$ at MSL and the temperatures at both MSL and at the given altitude ($t_0, t$).

$$\begin{equation} p = p_0(t/t_0)^{5.256} \label{eq:pressLapse} \end{equation}$$

ISA specifies $p_0 = 2116.4 lb/ft^2$ on a “standard day,” but again pilots and engineers often adjust $p_0$ to model a specific (nonstandard) day.

If interested, you can derive Equation \ref{eq:pressLapse} from more fundamental principles. This is done in equations 7-14 in Chapter 13 of Embry-Riddle’s “Introduction to Aerospace Flight Vehicles.”

Putting it together

Let’s put this together as would be seen in a helicopter simulation. Let’s assume the temperature and pressure at the initial altitude are given: $t_i, p_i, h_i$. As the helicopter flies around we’ll need to compute the temperature $t$, pressure $p$, and density $\rho$ at an arbitrary altitude.

The first thing we’ll do is compute and save the temperature and pressure at MSL. We know $t_i=t_0-\lambda h_i$ and therefore with a slight rearrangement we can compute the temperature $t_0$ at MSL as below.

$$\begin{equation} t_0 = t_i + \lambda h_i \label{eq:temp0} \end{equation}$$

Given these values, we can rearrange Equation \ref{eq:pressLapse} to compute the pressure $p_0$ at MSL as below.

$$\begin{equation} p_0 = \frac{p}{(t/t_0)^{5.256}} \label{eq:pres0} \end{equation}$$

Now that we have $t_0$ and $p_0$ we can compute the temperature and pressure at any altitude $h$ thereafter using Equitions \ref{eq:tempLapse}-\ref{eq:pressLapse}. With $t,p$ we can then compute the density $\rho$ using Equation \ref{eq:thermoState}: $\rho = p/Rt$.

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